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\begin{titlepage}
  \vspace*{-0cm}
  \begin{center}
    % Top vertical fill
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    % Course Name (largest)
    {\Huge MATH 223: Linear Algebra}\\[0.5cm]
    
    % University Name
    {\Large McGill University}\\[1.5cm]
    
    % Instructor and Notes by
    {\large Instructor: Prof.\ Jeremy Macdonald}\\[0.2cm]
    {\large Notes by: Cagri Arslan}\\[1.5cm]
    
    % Condensed Notes and Date
    {\large Condensed Notes}\\[0.2cm]
    {\large July 1, 2025}\\[2cm]
    
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      \draw[->] (V4) -- node[right] {$Q_{\alpha}^{\beta}$} (V2);
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    \vspace{0.3cm}
    {\footnotesize The Similarity Transform (Theorem 68)}

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  \Huge \textbf{Foreword}
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\vspace{1em}

% Write your foreword text here
\noindent
This document contains my condensed notes for Jeremy Macdonald's MATH 223 class, taken in W2025. These notes were typeset manually by myself, in their entirety.\\

\noindent
It is important to note that these are \textit{condensed notes}, and do not account for a comprehensive treatment of MATH 223. These notes do not provide proofs of theorems, nor do they contain any discussion pertaining to the content of the course. They simply list out all covered theorems, propositions, and definitions. Thus, simply reading this document is unlikely to be sufficient practice to succeed in MATH 223.\\

\noindent
For context, I had to take one of MATH 223 and MATH 247 for my Math minor. I chickened out, thinking MATH 223 was harder than it actually was, and took 223 instead of 247. Add-Drop has passed by the time I realized that I would be better served by 247, so I went to Prof. Tim Hoheisel's MATH 247 lectures, only attending 223 for exams. Since Hoheisel provided his own notes, I did not take any for 247. I wrote these up a few days before the 223 final, and proofread them in the summer. Babbling aside, the takeaway here is that you shouldn't be afraid to challenge yourself.\\

\noindent
These notes consist of five discrete sections, following Prof. Macdonald's course outline. Additionally, each statement is one of the following, and is labeled as such.

\begin{description}
  \item[\textbf{Definition:}] An explanation of the mathematical meaning of a word or phrase.
  \item[\textbf{Theorem:}] A significant statement that has been proven to be true.
  \item[\textbf{Proposition:}] A less significant, but nonetheless important, proven-true statement.
  \item[\textbf{Lemma:}] A simpler, true statement used in proving other true statements.
  \item[\textbf{Corollary:}] A true statement that is a simple deduction from a theorem or proposition.
\end{description}

\noindent
You can contact me through the following channels. My GitHub also contains notes for some other courses. I monitor both email accounts, but might not check the McGill one too often after I graduate.

\begin{itemize}
  \item \textbf{School e-mail:} 
    \href{mailto:cagri.arslan@mail.mcgill.ca}{cagri.arslan@mail.mcgill.ca}
  \item \textbf{Personal e-mail:} 
    \href{mailto:cagriarslan9@gmail.com}{cagriarslan9@gmail.com}
  \item \textbf{GitHub:} 
    \href{https://github.com/imported-canuck}{github.com/imported-canuck}
\end{itemize}

\noindent
\textit{This project is licensed under the MIT license. All content relating to this document is intended to be available to anyone, free of charge. I will \textbf{never} request payment for access to any portion of this document. If anyone attempts to sell you this, do not buy it. Instead, email me and I will send you the complete notes, free of charge.}
\clearpage

\section{Complex Numbers}

\subsection{Basics of \(\mathbb{C}\)}

\begin{defn}[Field of Complex Numbers]
The field of complex numbers, \(\mathbb{C}\), consists of all expressions of the form
\[
a + b\,i,
\]
where \(a,b\in\mathbb{R}\) and \(i\) is a symbol satisfying \(i^2=-1\).
\end{defn}

\begin{defn}[Addition and Multiplication]
Let \(z = a_1 + b_1\,i\) and \(\omega = a_2 + b_2\,i\).  Define
\[
z + \omega = (a_1 + a_2) + (b_1 + b_2)\,i,
\qquad
z\,\omega = (a_1 + b_1 i)(a_2 + b_2 i)
= (a_1a_2 - b_1b_2) + (a_1b_2 + a_2b_1)\,i.
\]
\end{defn}

\begin{defn}[Conjugate and Modulus]
For \(z = a + b\,i\),
\[
\overline{z} = a - b\,i
\quad\text{(complex conjugate)}, 
\qquad
|z| = \sqrt{a^2 + b^2}
\quad\text{(absolute value or modulus)}.
\]
\end{defn}

\begin{prop}
If \(z = a + b\,i \neq 0\) (i.e.\ \(z \neq 0 + 0i\)), then the number
\[
z^{-1} \;=\; \frac{\overline{z}}{\lvert z \rvert^2} 
\;=\; \frac{a}{a^2 + b^2} \;-\; \frac{b}{a^2 + b^2}\,i
\]
is called the \emph{(multiplicative) inverse} of \(z\) and satisfies 
\[
z \, z^{-1} \;=\; 1 
\;=\; z^{-1}\,z.
\]
\end{prop}

\begin{defn}[Division]
For \(z,\omega\in\mathbb{C}\) with \(\omega\neq0\), define
\[
\frac{z}{\omega} \;=\; z\,\omega^{-1}.
\]
\end{defn}

\begin{prop}
Let \(z,w \in \mathbb{C}\).  
\begin{enumerate}
\item \(\overline{z + w} = \overline{z} + \overline{w}\).
\item \(\overline{z\,w} = \overline{z}\,\overline{w}\).
\item \(\overline{\,\overline{z}\,} = z\).
\item \(z \,\overline{z} = \lvert z\rvert^{2}\).
\item \(z \in \mathbb{R} \;\Longleftrightarrow\; \overline{z} = z\).
\end{enumerate}
\end{prop}

\subsection{Polar Form and Equation}

\begin{defn}[Complex Exponential Function]
In the complex plane, \(e^{i\theta}\) is the unique number of modulus 1 and argument \(\theta\).  Equivalently,
\[
e^{i\theta} = \cos\theta + i\sin\theta,
\]
(Euler’s formula).
\end{defn}

\begin{thm}[Fundamental Theorem of Algebra]
Let 
\[
p(z) \;=\; a_n\,z^n \;+\; a_{n-1}\,z^{n-1} \;+\;\cdots+\; a_1\,z \;+\; a_0,
\]
where \(a_i \in \mathbb{C}\). Then \(p(z)\) factors into linear factors:
\[
p(z) \;=\; a_n \,(z - r_1)\,(z - r_2)\,\cdots\,(z - r_n),
\]
where each \(r_i \in \mathbb{C}\) (the \(r_i\) may repeat).
\end{thm}

\section{Vector Spaces}

\subsection{Vector Space Axioms}

\begin{defn}[Field]
A \emph{field} is a set \(F\) equipped with two operations \(+\) and \(\cdot\) such that:
\begin{itemize}
  \item \((F,+)\) is an abelian group.
  \item \((F\setminus\{0\},\cdot)\) is an abelian group.
  \item Multiplication distributes over addition: for all \(a,b,c\in F\),
    \[
      a\,(b+c)=ab+ac,
      \qquad
      (a+b)\,c=ac+bc.
    \]
\end{itemize}
Examples include \(\mathbb{Q},\mathbb{R},\mathbb{C}\).  \(\mathbb{Z}\) is \emph{not} a field since, e.g., \(\tfrac23\notin\mathbb{Z}\).
\end{defn}

\begin{defn}[Vector Space]
Let \(F\) be a field and \(V\) a set.  A \emph{vector space over \(F\)} is a pair \((V,+)\) together with a scalar multiplication \(F\times V\to V\) satisfying the following eight axioms for all \(u,v,w\in V\) and \(a,b\in F\):
\begin{enumerate}
  \item \(u+v = v+u\).  
  \item \((u+v)+w = u+(v+w)\).  
  \item There exists \(0\in V\) with \(u+0=0+u=u\).  
  \item For each \(u\) there is \(-u\) with \(u+(-u)=(-u)+u=0\).  
  \item \(a\,(u+v)=au + av\).  
  \item \((a+b)\,u = au + bu\).  
  \item \(a\,(b\,u) = (ab)\,u\).  
  \item \(1\,u = u\), where \(1\) is the multiplicative identity in \(F\).
\end{enumerate}
\end{defn}


\begin{prop}[Basic Vector Space Properties]
Let \(V\) be a vector space over a field \(F\).
\begin{enumerate}
\item For all \(u,v,w \in V\), if \(u + w = v + w\) then \(u = v\).
\item The zero vector \(\vec{0}\) in \(V\) is unique.
\item For each \(u \in V\), its additive inverse \(-u\) is unique.
\item For all \(u \in V\), \(0\,u = \vec{0}\).
\item For all \(c \in F\), \(c\,\vec{0} = \vec{0}\).
\item For all \(c \in F\) and \(u \in V\), \((-c)u = c(-u) = -(cu)\).
\end{enumerate}
\end{prop}

\subsection{Linear Combinations and Subspaces}

\begin{defn}[Linear Combination]
Given vectors \(u_1,u_2,\dots,u_m\in V\) and scalars \(c_1,\dots,c_m\in F\), any vector of the form
\[
  c_1\,u_1 + c_2\,u_2 + \cdots + c_m\,u_m
\]
is called a \emph{linear combination} of \(u_1,\dots,u_m\).
\end{defn}

\begin{defn}[Span]
Let \(S=\{u_1,u_2,\dots,u_m\}\subseteq V\).  The \emph{span} of \(S\) is
\[
  \operatorname{span}(S)
  \;=\;
  \bigl\{\,c_1u_1 + c_2u_2 + \cdots + c_mu_m
  \;\big|\;
  c_i\in F\bigr\}.
\]
If \(S=\varnothing\), we define \(\operatorname{span}(\varnothing)=\{0\}\).
\end{defn}

\begin{prop}
If \(A,B \in M_{m \times n}(F)\) and \(B\) is obtained from \(A\) by elementary row operations (EROs), then
\[
\mathrm{row}(A) = \mathrm{row}(B).
\]
\end{prop}

\begin{prop}[Facts About Spans]
Let \(S \subseteq V\). Then:
\begin{enumerate}
\item For all \(u,w \in \mathrm{span}(S)\), \(u + w \in \mathrm{span}(S)\) 
  \quad(\emph{closure under addition}).
\item For all \(u \in \mathrm{span}(S)\) and \(c \in F\), \(cu \in \mathrm{span}(S)\) 
  \quad(\emph{closure under scalar multiplication}).
\item \(\vec{0} \in \mathrm{span}(S)\).
\end{enumerate}
\end{prop}

\begin{defn}[Subspace]
Let \(V\) be a vector space over a field \(F\), and let \(W\subseteq V\).  We say \(W\) is a \emph{subspace} of \(V\) (and write \(W \leq V\)) if:
\begin{enumerate}
  \item For all \(w_1,w_2\in W\), \(w_1+w_2\in W\).
  \item For all \(w\in W\) and all scalars \(c\in F\), \(c\,w\in W\).
  \item The zero vector \(0\in W\).
\end{enumerate}
\end{defn}

\begin{thm}
Let \(A \in M_{m \times n}(F)\), \(b \in F^m\), and let \(x \in F^n\) be the vector of variables.  
Let \(S\) be the set of all solutions to the linear system \(A\,x = b\).  
Then \(S\) is a subspace of \(F^n\) if and only if 
\[
b = 
\begin{pmatrix}
0 \\ \vdots \\ 0
\end{pmatrix} 
= \vec{0}
\quad (\text{i.e.\ the system is homogeneous}).
\]
\end{thm}

\begin{prop}
Subspaces are closed under forming linear combinations.  
If \(W \le V\), then for any positive integer \(n\), if \(w_1,w_2,\dots,w_n \in W\) and \(c_1,\dots,c_n \in F\), then
\[
c_1\,w_1 + c_2\,w_2 + \cdots + c_n\,w_n \;\in\; W.
\]
\end{prop}

\begin{prop}[Spans Are Subspaces]
Let \(S \subseteq V\) (a subset) and \(W \le V\) (a subspace). Then:
\begin{enumerate}
\item \(S \subseteq \mathrm{span}(S)\).
\item If \(S \subseteq W\), then \(\mathrm{span}(S) \subseteq W\).
\item \(\mathrm{span}(W) = W\).
\end{enumerate}
\end{prop}

\subsection{Linear Independence and Dependence}

\begin{defn}[Linear Dependence]
Let \(V\) be a vector space over \(F\) and \(S\subseteq V\).  The set \(S\) is \emph{linearly dependent} if there exist distinct vectors \(u_1,\dots,u_n\in S\) and scalars \(c_1,\dots,c_n\in F\), not all zero, such that
\[
c_1u_1 + c_2u_2 + \cdots + c_nu_n \;=\; 0.
\]
Otherwise, \(S\) is \emph{linearly independent}.
\end{defn}

\begin{defn}[Linear Independence]
Let \(V\) be a vector space over a field \(F\) and let \(S\subseteq V\).  The set \(S\) is said to be \emph{linearly independent} if whenever distinct vectors \(u_1,\dots,u_n\in S\) and scalars \(c_1,\dots,c_n\in F\) satisfy
\[
c_1u_1 + c_2u_2 + \cdots + c_nu_n \;=\; 0,
\]
it follows that \(c_1 = c_2 = \cdots = c_n = 0\).
\end{defn}

\begin{prop}[Dependency Special Cases]\leavevmode\vspace{-20.5pt}
\begin{enumerate}
\item The empty set \(\varnothing\) is linearly independent.
\item Let \(S \subseteq V\). If \(\vec{0} \in S\), then \(S\) is dependent 
  (since \(1 \cdot \vec{0} = \vec{0}\) provides a nontrivial dependence).
\item Let \(u \in V\). Then \(\{u\}\) is independent if and only if \(u \neq \vec{0}\).  
  Equivalently, \(\{u\}\) is dependent if and only if \(u = \vec{0}\).
\item Let \(A \subseteq B \subseteq V\). Then:
  \begin{enumerate}
  \item If \(A\) is dependent, then \(B\) is also dependent.
  \item If \(B\) is independent, then \(A\) is also independent.
  \end{enumerate}
\end{enumerate}
\end{prop}

\begin{lem}[Extending a Linearly Independent Set]
Let \(S \subseteq V\) be a linearly independent set, and let \(w \in V\) with \(w \notin S\).  
Then \(S \cup \{w\}\) is independent if and only if \(w \notin \mathrm{span}(S)\).  
(Adding a vector already in the span of \(S\) makes the set dependent.)
\end{lem}

\subsection{Basis and Dimension}

\begin{defn}[Basis]
Let \(V\) be a vector space over \(F\) and let \(W\leq V\).  A subset \(\beta\subseteq W\) is called a \emph{basis} of \(W\) if
\begin{enumerate}
  \item \(\displaystyle \operatorname{span}(\beta) = W,\)
  \item \(\beta\) is linearly independent.
\end{enumerate}
\end{defn}

\begin{thm}["Bases Exist"]
Let \(W \le V\), and suppose \(W = \mathrm{span}(S)\) for some \emph{finite} set \(S\).  
Then there exists a subset \(\beta \subseteq S\) such that \(\beta\) is a basis of \(W\).  
(Any finite spanning set can be reduced to a basis.)
\end{thm}

\begin{thm}["All Bases Have the Same Size"]
Let \(W = \mathrm{span}(S)\) with \(S\) finite. Then \(W\) has a finite basis, and \emph{all} bases of \(W\) have the same cardinality. This common number is called \(\dim(W)\).
\end{thm}

\begin{defn}[Dimension]
A vector space \(V\) (or subspace of one) is called \emph{finite–dimensional} if it admits a finite basis.  The \emph{dimension} of \(V\), written \(\dim(V)\), is the number of vectors in any basis of \(V\).  If no finite basis exists, then \(V\) is \emph{infinite–dimensional}.
\end{defn}

\begin{thm}
Every vector space (even one without a finite spanning set) has a basis.
\end{thm}

\begin{prop}
If we find the general solution to \(A\,\vec{x} = \vec{0}\) as 
\[
\vec{x} \;=\; t_1\,\vec{v}_1 \;+\; t_2\,\vec{v}_2 \;+\;\cdots+\; t_r\,\vec{v}_r,
\]
where \(t_1,\dots,t_r\) are the free variables, then \(\{\vec{v}_1,\vec{v}_2,\dots,\vec{v}_r\}\) forms a basis for \(\mathrm{null}(A)\).
\end{prop}

\begin{thm}
Suppose \(\dim(W) = n\) is finite, and let \(S \subseteq W\).
\begin{enumerate}
\item If \(W = \mathrm{span}(S)\), then \(\lvert S\rvert \ge n\), and there is a subset \(\beta \subseteq S\) such that \(\beta\) is a basis of \(W\).  
(Any spanning set can be ``shrunk'' to a basis.)
\item If \(S\) is independent, then \(\lvert S\rvert \le n\), and there is a basis \(\beta\) of \(W\) such that \(S \subseteq \beta\).  
(Any independent set can be ``extended'' to a basis.)
\item Let \(\lvert S\rvert = n\). If and only if \(W = \mathrm{span}(S)\), then \(S\) is independent.
\end{enumerate}
\end{thm}

\subsection{Subspaces}

\begin{prop}
Let \(W \leq V\), where \(\dim(V) = n\) is finite.
\begin{enumerate}
\item \(\dim(W) \le n.\)
\item \(\dim(W) = n \;\Longleftrightarrow\; W = V.\)
\end{enumerate}
\end{prop}

\begin{defn}[Column Space]
Let \(A\in M_{m\times n}(F)\).  The \emph{column space} of \(A\), denoted \(\mathrm{col}(A)\), is the subspace of \(F^m\) spanned by the columns of \(A\):
\[
\mathrm{col}(A)
=\mathrm{span}\{\text{columns of }A\}
\;\le F^m.
\]
\end{defn}

\begin{thm}
Let \(A \in M_{m\times n}(F)\), and let \(R\) be the row-reduced echelon form of \(A\).  
\begin{enumerate}
\item A basis for \(\mathrm{row}(A)\) is given by the nonzero rows of \(R\).
\item The columns of \(A\) that correspond to the leading entries (pivots) in \(R\) form a basis for \(\mathrm{col}(A)\).
\end{enumerate}
\end{thm}

\begin{thm}
Let \(U \le V\) and \(W \le V\) be subspaces of \(V\). Then 
\[
U \,\cap\, W \;=\; \{\,v \in V : v \in U \text{ and } v \in W\}
\]
is a subspace of \(V\).
\end{thm}

\begin{defn}[Sum of Subspaces]
Let \(V\) be a vector space over a field \(F\) and let \(U,W\leq V\).  The \emph{sum} of \(U\) and \(W\) is the subspace
\[
U + W \;=\;\{\,u + w \mid u\in U,\;w\in W\}\;\leq V.
\]
\end{defn}

\begin{prop}
Let \(U,W\) be subspaces of \(V\). Then:
\begin{enumerate}
\item \(U + W = \mathrm{span}(U \cup W)\).
\item \(U \le U + W\) and \(W \le U + W\).
\end{enumerate}
\end{prop}

\begin{defn}[Direct Sum]
Suppose \(U,W\leq V\) are subspaces such that every \(v\in V\) can be written uniquely as
\[
v = u + w
\quad\text{with }u\in U,\;w\in W.
\]
Then \(V\) is called the \emph{direct sum} of \(U\) and \(W\), and we write
\[
V = U \oplus W.
\]
\end{defn}

\begin{prop}
Let \(U,W\) be subspaces of \(V\). Then
\[
V = U \oplus W 
\;\Longleftrightarrow\;
\bigl(V = U + W\bigr) \text{ and } \bigl(U \,\cap\, W = \{\vec{0}\}\bigr).
\]
\end{prop}

\begin{thm}[Inclusion-Exclusion Theorem]
Let \(U,W\) be finite-dimensional subspaces of \(V\). Then
\[
\dim(U + W) \;=\; \dim(U)\;+\;\dim(W)\;-\;\dim\bigl(U \,\cap\, W\bigr).
\]
\end{thm}

\subsection{Lagrange Interpolation}

\begin{defn}[Lagrange Polynomials]
Let $a_0,\dots,a_n\in\mathbb{R}$ be distinct.  For each $i=0,1,\dots,n$, the \emph{Lagrange polynomial} $\ell_i(x)$ is defined by
\[
\ell_i(x)\;=\;\prod_{\substack{0\le j\le n\\j\neq i}}
\frac{x - a_j}{a_i - a_j}.
\]
Each $\ell_i(x)$ has degree $n$ and satisfies $\ell_i(a_j)=\delta_{ij}$.
\end{defn}

\begin{defn}[Kronecker Delta]
The \emph{Kronecker delta} $\delta_{ij}$ is defined for any indices $i,j$ by
\[
\delta_{ij}\;=\;
\begin{cases}
1,&i=j,\\
0,&i\neq j.
\end{cases}
\]
\end{defn}

\begin{prop}\leavevmode\vspace{-20.5pt}
\begin{enumerate}
\item If \(l_i(a_j) = \delta_{i,j}\), then \(\delta_{i,j} = 
\begin{cases}
1 & \text{if } j = i,\\
0 & \text{if } j \neq i,
\end{cases}\) (Kronecker delta).
\item The polynomials \(l_0(x),\,l_1(x),\dots,\,l_n(x)\) form a basis of \(P_n(\mathbb{R})\).
\end{enumerate}
\end{prop}

\section{Linear Transformations}

\subsection{Definition and Basic Properties}

\begin{defn}[Linear Transformation]
Let $U$ and $V$ be vector spaces over a field $F$, and let $T\colon U\to V$ be a function.  If for all $u_1,u_2\in U$ and all $c\in F$,
\[
T(u_1+u_2)=T(u_1)+T(u_2),
\qquad
T(c\,u)=c\,T(u),
\]
then $T$ is called a \emph{linear transformation}.
\end{defn}

\begin{prop}[Properties of Linear Transformations]
Let \(T : U \to V\) be a linear transformation. Then:
\begin{enumerate}
\item \(T(\vec{0}) = \vec{0}\).
\item For all \(u_1, \dots, u_n \in U\) and \(c_1, \dots, c_n \in F\),
\[
T\Bigl(\sum_{i=1}^n c_i u_i\Bigr) 
\;=\; \sum_{i=1}^n c_i \, T(u_i).
\]
\end{enumerate}
\end{prop}

\begin{defn}[Matrix‐Induced Linear Map]
For any $A\in M_{m\times n}(F)$, define the map
\[
L_A\colon F^n \;\longrightarrow\; F^m,
\qquad
L_A(v)=A\,v,
\]
for all $v\in F^n$.  One checks that $L_A$ is linear.
\end{defn}

\begin{prop}
\(L_A\) is a linear transformation.
\end{prop}

\subsection{Kernel and Image}

\begin{defn}[Kernel and Image]
Let $T\colon U\to V$ be a linear transformation.  The \emph{kernel} of $T$ is
\[
\ker(T)=\{\,u\in U\mid T(u)=0\}\subseteq U,
\]
and the \emph{image} of $T$ is
\[
\operatorname{Im}(T)=\{\,v\in V\mid v=T(u)\text{ for some }u\in U\}\subseteq V.
\]
\end{defn}

\begin{prop}
Let \(T : U \to V\) be a linear transformation. Then:
\begin{enumerate}
\item \(\ker(T) \le U\) (i.e., \(\ker(T)\) is a subspace of \(U\)).
\item \(\mathrm{Im}(T) \le V\) (i.e., \(\mathrm{Im}(T)\) is a subspace of \(V\)).
\end{enumerate}
\end{prop}

\begin{defn}[Rank and Nullity]
Let $T\colon U\to V$ be a linear transformation between finite‐dimensional spaces.  Then
\[
\mathrm{rank}(T)=\dim\bigl(\operatorname{Im}(T)\bigr),
\qquad
\mathrm{nullity}(T)=\dim\bigl(\ker(T)\bigr).
\]
\end{defn}

\begin{prop}[Spanning Set of $\mathrm{Im}(T)$]
\(T: U \to V\) linear, \(U = \mathrm{span}(\alpha)\).

Denote \(T(\alpha) = \{\,T(u) \mid u \in \alpha\}\).  
Then \(T(\alpha)\) spans \(\mathrm{Im}(T)\).
\end{prop}

\begin{thm}[Rank-Nullity Theorem]
\(T : U \to V\) linear transformation, with \(\dim(U) = n\) finite.  
Then 
\[
\mathrm{rank}(T) \;+\; \mathrm{nullity}(T) \;=\; \dim(U).
\]
\end{thm}

\subsection{Injective, Surjective}

\begin{defn}[Injective, Surjective, Bijective]
Let \(f: X \to Y\) be a function between sets \(X\) and \(Y\).
\begin{itemize}
  \item \(f\) is \emph{injective} (one-to-one) if
    \[
      \forall\,x_1,x_2\in X,\;x_1\neq x_2\;\Longrightarrow\;f(x_1)\neq f(x_2),
    \]
    equivalently \(\;f(x_1)=f(x_2)\implies x_1=x_2.\)
  \item \(f\) is \emph{surjective} (onto) if
    \[
      \forall\,y\in Y\;\exists\,x\in X\text{ with }f(x)=y,
    \]
    equivalently \(\mathrm{Im}(f)=Y.\)
  \item \(f\) is \emph{bijective} if it is both injective and surjective.
\end{itemize}
\end{defn}

\begin{prop}
\(T : U \to V\) linear, \(U, V\) finite dimensional.  
\begin{enumerate}
\item \(T\) injective \(\Longleftrightarrow\) \(\ker(T) = \{\vec{0}\}\)  
  (\(\Longrightarrow\) \(\mathrm{nullity}(T) = 0\)).
\item \(T\) surjective \(\Longleftrightarrow\) \(\mathrm{Im}(T) = V\)  
  (\(\Longrightarrow\) \(\mathrm{rank}(T) = \dim(V)\)).
\item If \(\dim(U) = \dim(V)\), then \(T\) injective \(\Longleftrightarrow\) \(T\) surjective.
\item If \(\dim(U) > \dim(V)\), \(T\) is not injective.  
  If \(\dim(U) < \dim(V)\), \(T\) is not surjective.
\end{enumerate}
\end{prop}

\subsection{Isomorphism and Coordinates}

\begin{defn}[Isomorphism of Vector Spaces]
Let \(T:U\to V\) be a linear transformation between vector spaces over the same field.
\begin{itemize}
  \item If \(T\) is bijective, then \(T\) is called an \emph{isomorphism}.
  \item When such an isomorphism \(T\) exists, we say \(U\) is \emph{isomorphic} to \(V\) and write
    \[
      U \cong V.
    \]
\end{itemize}
\end{defn}

\begin{prop}
Let \(\beta = \{\vec{v}_1, \dots, \vec{v}_n\}\) be a basis of \(V\). Then every \(\vec{v} \in V\) has a 
\emph{unique} expression
\[
\vec{v} \;=\; \sum_{i=1}^n c_i \,\vec{v}_i
\]
as a linear combination of basis elements. The vector \((c_1, c_2, \dots, c_n) \in F^n\) of coefficients 
is called the \emph{coordinate vector} of \(\vec{v}\) relative to \(\beta\), denoted 
\(\,[\vec{v}]_\beta = (c_1, \dots, c_n).\)
\end{prop}

\begin{thm}
Let \(V\) and \(W\) be vector spaces, let \(\alpha = \{v_1,\dots,v_n\}\) be a basis of \(V\), and let \(w_1,\dots,w_n\in W\) be arbitrary.  Then there is a unique linear transformation \(T:V\to W\) such that
\[
T(v_i) = w_i\quad (i=1,\dots,n).
\]
Moreover, if 
\(\displaystyle v = \sum_{i=1}^n c_i\,v_i,\) 
then 
\[
T(v) \;=\; \sum_{i=1}^n c_i\,w_i.
\]
\end{thm}

\begin{cor}
If \(T,S:V\to W\) are linear and \(\alpha=\{v_1,\dots,v_n\}\) is a basis of \(V\) with
\[
T(v_i) = S(v_i)\quad\text{for }i=1,\dots,n,
\]
then \(T=S\).
\end{cor}

\begin{thm}
\(V\) vector space, \(\dim V = n\), finite, \(\beta\) basis.  
The function which computes coordinates 
\[
[-]_\beta : V \to F^n
\]
is an isomorphism. Hence \(V \cong F^n\) \quad \((n = \dim V)\).
\end{thm}

\begin{prop}[Composition and Inverses of Lin. Transforms]
Let \(S: U \to V\), \(T: V \to W\) be linear.
\begin{enumerate}
\item The composition \(T \circ S: U \to W\) is linear.
\item If \(S, T\) both isomorphisms, \(T \circ S\) also is an isomorphism.
\item If \(T\) is an isomorphism, there is an inverse \(T^{-1}: W \to V\), and \(T^{-1}\) is an isomorphism.
\end{enumerate}
\end{prop}

\subsection{Matrix of a Linear Transformation}

\begin{thm}[“Only Dimensions Matter”]
Let \(U, V\) be finite-dimensional vector spaces over \(F\). Then 
\[
U \cong V \quad (\Longleftrightarrow)\quad \dim(U) = \dim(V).
\]
\end{thm}

\begin{defn}[Matrix of a Linear Transformation]
Let \(T\colon U\to V\) be a linear transformation, and let 
\(\alpha=\{u_1,\dots,u_n\}\) and \(\beta=\{v_1,\dots,v_m\}\)
be ordered bases of \(U\) and \(V\), respectively.  The \emph{matrix of \(T\) relative to \(\alpha\) and \(\beta\)} is the \(m\times n\) matrix
\[
[T]_{\alpha}^{\beta}
=
\bigl([T(u_1)]_{\beta}\;\big|\;\dots\;\big|\;[T(u_n)]_{\beta}\bigr),
\]
whose \(i\)th column is the coordinate vector of \(T(u_i)\) in the basis \(\beta\).
\end{defn}

\begin{thm}["{\([T]_\alpha^\beta\) Computes \(T\) in Coordinates}"]
Let \(T: U \to V\) be linear, \(\alpha, \beta\) bases of \(U, V\). Then for all \(u \in U\),
\[
[T]_\alpha^\beta \,[u]_\alpha \;=\; [\,T(u)\,]_\beta.
\]
\end{thm}

\begin{prop}[{$\ker$ and $\mathrm{Im}$ in Coordinates}]
Let \(T : U \to V\), and let \(\alpha, \beta\) be bases of \(U\) and \(V\). Define \(A = [T]_{\alpha}^{\beta}\).
\end{prop}

\begin{enumerate}
\item \(\ker(T)\) corresponds, via \(\alpha\)-coordinates, to \(\mathrm{null}(A)\) (i.e.\ the solution set of \(A \vec{x} = \vec{0}\)). In particular, 
\[
\mathrm{nullity}(T) \;=\; \mathrm{nullity}(A) \;=\; \#\{\text{free variables}\}.
\]

\item \(\mathrm{Im}(T)\) corresponds, via \(\beta\)-coordinates, to \(\mathrm{Im}(L_A) = \mathrm{col}(A)\). Hence
\[
\mathrm{rank}(T) \;=\; \mathrm{rank}(A).
\]
\end{enumerate}

\begin{defn}[Linear Operator]
Let \(T\colon V\to V\) be a linear transformation on a vector space \(V\) (the domain and codomain are the same vector space).  Such a \(T\) is called a \emph{linear operator}.  If \(\dim(V)=n\), then its matrix relative to any basis \(\alpha\) of \(V\) is an \(n\times n\) matrix, commonly denoted
\[
[T]_{\alpha}^{\alpha} = [T]_{\alpha},
\]
or simply \([T]\) when the basis is understood.
\end{defn}

\begin{prop}
Let \(T: V \to V\) be a linear operator, and let \(\alpha\) be a basis of \(V\) with \(n = \dim(V)\).  Then
\[
T\text{ is invertible}
\quad\Longleftrightarrow\quad
[T]_{\alpha}\text{ is invertible}.
\]
Moreover, if \(T\) is invertible, then
\[
\bigl[T^{-1}\bigr]_{\alpha}
\;=\;
\bigl([T]_{\alpha}\bigr)^{-1}.
\]
\end{prop}

\subsection{Change of Basis}

\begin{defn}[Change‐of‐Coordinates Matrix]
Let \(V\) be an \(n\)-dimensional vector space over a field \(F\), and let 
\(\alpha\) and \(\beta\) be two ordered bases of \(V\).  The \emph{change‐of‐coordinates matrix from \(\alpha\) to \(\beta\)} is the \(n\times n\) matrix
\[
Q^\beta_\alpha \;=\; [I]^{\beta}_{\alpha},
\]
where \(I:V\to V\) is the identity.  Equivalently, the \(j\)th column of \(Q^\beta_\alpha\) is the coordinate vector of the \(j\)th basis vector of \(\alpha\) expressed in the basis \(\beta\).
\end{defn}

\begin{prop}\leavevmode\vspace{-20.5pt}
\begin{enumerate}
\item For every $u\in V$,
\[
Q^\beta_\alpha\,[u]_\alpha \;=\; [u]_\beta.
\]
\item $Q^\beta_\alpha$ is invertible and 
\[
\bigl(Q^\beta_\alpha\bigr)^{-1} \;=\; Q^\alpha_\beta.
\]
\end{enumerate}
\end{prop}

\begin{thm}[Similarity Transformation]
Let $T:V\to V$ be a linear operator and let $\alpha,\beta$ be two bases of $V$.  Then
\[
Q^\beta_\alpha \,[T]_\alpha \,Q^\alpha_\beta \;=\; [T]_\beta.
\]
\end{thm}

\begin{defn}[Similar Matrices]
Let \(A,B\in M_{n\times n}(F)\).  We say \(A\) and \(B\) are \emph{similar} if there exists an invertible matrix \(Q\in M_{n\times n}(F)\) such that
\[
Q^{-1} \,A\,Q \;=\; B.
\]
\end{defn}

\begin{prop}
Let $Q\in M_{n\times n}(F)$ be any invertible matrix and let $\alpha$ be a basis of $V$ (so $n=\dim V$).  Then there exists a basis $\beta$ of $V$ such that
\[
Q \;=\; Q^\beta_\alpha.
\]
\end{prop}

\begin{prop}
Let $T:V\to V$ be a linear operator, let $\alpha$ be a basis of $V$, and let $B\in M_{n\times n}(F)$ be any matrix.  Then
\[
[T]_\alpha \text{ is similar to } B
\quad\Longleftrightarrow\quad
\exists \text{ a basis }\beta\text{ of }V\text{ with }B = [T]_\beta.
\]
\end{prop}

\begin{thm}
If $\dim V = n$ and $\dim W = m$, then
\[
\mathcal{L}(V,W)\;\cong\;M_{m\times n}(F).
\]
Moreover, given a basis $\alpha$ of $V$ and a basis $\beta$ of $W$, the map
\[
\varphi:\mathcal{L}(V,W)\;\longrightarrow\;M_{m\times n}(F),
\qquad
\varphi(T) = [T]_\alpha^\beta
\]
is a vector‐space isomorphism.
\end{thm}

\section{Inner Product Spaces}

\subsection{Definition and Main Examples}

\begin{defn}[Inner Product]
Let \(V\) be a vector space over \(\mathbb{F}\) (where \(\mathbb{F}=\mathbb{R}\) or \(\mathbb{C}\)).  An \emph{inner product} on \(V\) is a function
\[
\langle\,\cdot\,,\,\cdot\,\rangle \colon V\times V \;\longrightarrow\; \mathbb{F}
\]
satisfying for all \(u,v,w\in V\) and all scalars \(c\in\mathbb{F}\):
\begin{enumerate}
  \item \(\displaystyle \langle u,\,v+w\rangle = \langle u,\,v\rangle + \langle u,\,w\rangle,\)
    \quad\(\langle u,\,c\,v\rangle = c\,\langle u,\,v\rangle.\)
    \quad(\emph{linearity}).
  \item \(\displaystyle \langle u,\,v\rangle = \overline{\langle v,\,u\rangle}\)
    \quad(\emph{conjugate symmetry}).
  \item \(\displaystyle \langle u,\,u\rangle > 0\) for all \(u\neq 0\)
    \quad(\emph{positive definiteness}).
\end{enumerate}
\end{defn}

\begin{prop}[Inner Product Properties]
Let $V$ be an inner‐product space over $\mathbb F$ and let $u,v,w\in V$, $c\in\mathbb F$.  Then:
\begin{enumerate}
\item \emph{(Conjugate‐linearity in the second slot)}  
\[
\langle u,\,v+w\rangle = \langle u,v\rangle + \langle u,w\rangle,
\qquad
\langle u,\,c\,v\rangle = \overline{c}\,\langle u,v\rangle.
\]
\item $\langle u,0\rangle = 0$ and $\langle 0,u\rangle = 0$.
\item $\langle u,u\rangle = 0$ if and only if $u=0$.
\item If $\langle u,w\rangle = \langle v,w\rangle$ for all $w\in V$, then $u=v$.
\end{enumerate}
\end{prop}

\begin{defn}[Conjugate and Adjoint of a Matrix]
Let \(A = (a_{ij})\in M_{m\times n}(\mathbb{F})\).  
\begin{itemize}
  \item The \emph{conjugate} of \(A\) is the entrywise–conjugated matrix
    \[
      \overline{A} \;=\;\bigl(\overline{a_{ij}}\bigr)
      \;\in\;M_{m\times n}(\mathbb{F}).
    \]
  \item If \(A\) is square (\(n=m\)), the \emph{adjoint} (or Hermitian transpose) of \(A\) is
    \[
      A^* \;=\;\overline{A}^{\,T}
      \;=\;\bigl(\overline{A}\bigr)^{T}
      \;\in\;M_{n\times n}(\mathbb{F}),
    \]
    characterized by \(\langle A\,u,\,v\rangle = \langle u,\,A^*v\rangle\) for all \(u,v\).
\end{itemize}
\end{defn}

\subsection{Norm and Angle}

\begin{defn}[Norm]
Let \(V\) be an inner‐product space with inner product \(\langle\cdot,\cdot\rangle\).  The \emph{norm} (or length) of a vector \(u\in V\) is
\[
\|u\|
\;=\;
\sqrt{\langle u,\,u\rangle}\,.
\]
\end{defn}

\begin{prop}
Let $V$ be an inner‐product space, $u\in V$, and $c\in\mathbb F$.  Then
\[
\|c\,u\| \;=\; |c|\,\|u\|.
\]
\end{prop}

\begin{thm}[Cauchy–Schwarz Inequality]
Let \(V\) be an inner-product space and \(u,v\in V\). Then
\begin{enumerate}
  \item \(\displaystyle \bigl\lvert\langle u,v\rangle\bigr\rvert \le \|u\|\,\|v\|\).
  \item Equality holds, \(\lvert\langle u,v\rangle\rvert = \|u\|\|v\|\), if and only if 
  \(u = c\,v\) (equivalently \(v = c\,u\)) for some scalar \(c\in\mathbb F\).
\end{enumerate}
\end{thm}

\begin{prop}[Triangle Inequality]
In any inner-product space \(V\), for all \(u,v\in V\),
\[
\|u+v\|\;\le\;\|u\| + \|v\|.
\]
\end{prop}

\begin{defn}[Angle]
Let \(V\) be a real inner‐product space and let \(u,v\in V\) be nonzero.  By the Cauchy–Schwarz inequality, the quotient
\[
\frac{\langle u,v\rangle}{\|u\|\;\|v\|}
\]
lies in \([-1,1]\), so there is a unique \(\theta\in[0,\pi]\) with
\[
\cos\theta
\;=\;
\frac{\langle u,v\rangle}{\|u\|\;\|v\|}.
\]
This \(\theta\) is called the \emph{angle} between \(u\) and \(v\).
\end{defn}

\subsection{Orthogonal Sets and Complements}

\begin{defn}[Orthogonal and Orthonormal Sets]
Let \(V\) be an inner‐product space.
\begin{enumerate}
  \item Two vectors \(u,v\in V\) are called \emph{orthogonal} if \(\langle u,v\rangle=0\).  (In particular, the zero‐vector is orthogonal to every vector.)
  \item A subset \(X\subseteq V\) is \emph{orthogonal} if every pair of distinct vectors in \(X\) is orthogonal and \(0\notin X\).
  \item If, in addition, \(\|u\|=1\) for all \(u\in X\), then \(X\) is \emph{orthonormal}.
  \item An orthonormal subset that is also a basis of \(V\) is called an \emph{orthonormal basis} (ONB).
\end{enumerate}
\end{defn}

\begin{prop}
Let \(V\) be an inner-product space and let \(X\subseteq V\) be an orthogonal set.  Then \(X\) is linearly independent.
\end{prop}

\begin{thm}[Fourier Coefficients]
Let \(\alpha=\{v_1,\dots,v_n\}\) be an orthogonal basis of the inner-product space \(V\).  Then for every \(u\in V\),
\[
u \;=\; \sum_{i=1}^n \frac{\langle u,v_i\rangle}{\langle v_i,v_i\rangle}\,v_i,
\]
and the coordinate vector \([u]_\alpha\) is
\[
[u]_\alpha
=
\begin{pmatrix}
\displaystyle\frac{\langle u,v_1\rangle}{\langle v_1,v_1\rangle}\\[6pt]
\vdots\\[3pt]
\displaystyle\frac{\langle u,v_n\rangle}{\langle v_n,v_n\rangle}
\end{pmatrix}.
\]
If \(\alpha\) is orthonormal, then this simplifies to
\[
u = \sum_{i=1}^n \langle u,v_i\rangle\,v_i.
\]
\end{thm}

\begin{defn}[Orthogonal Complement]
Let \(V\) be an inner‐product space and \(X\subseteq V\).  The \emph{orthogonal complement} of \(X\) is
\[
X^{\perp}
\;=\;
\{\,v\in V \mid \langle v,x\rangle = 0 \ \text{for all }x\in X\},
\]
which is a subspace of \(V\).
\end{defn}

\begin{prop}
Let \(W\le V\).  Then:
\begin{enumerate}
  \item \(W^\perp\) is a subspace of \(V\).
  \item If \(\alpha=\{w_1,\dots,w_k\}\) is a basis of \(W\), then for any \(u\in V\),
  \[
    u\in W^\perp
    \;\Longleftrightarrow\;
    \langle u,w_i\rangle = 0\quad (i=1,\dots,k).
  \]
  \item \(W\cap W^\perp = \{0\}.\)
\end{enumerate}
\end{prop}

\subsection{Orthogonal Projection and Gram-Schmidt Algorithm}

\begin{thm}[Orthogonal Projection]
Let \(V\) be an inner-product space and \(W\le V\) a finite-dimensional subspace.  Then:
\begin{enumerate}
  \item For each \(u\in V\) there are unique \(w\in W\) and \(w'\in W^\perp\) such that 
  \(\;u = w + w'\).  Hence \(V = W\oplus W^\perp\).
  \item If \(\{u_1,\dots,u_k\}\) is an orthogonal basis of \(W\), then the projection \(w\in W\) of any \(u\in V\) is
  \[
    w \;=\; \sum_{i=1}^k \frac{\langle u,u_i\rangle}{\langle u_i,u_i\rangle}\,u_i.
  \]
  \item If \(V\) is finite-dimensional, then 
  \(\dim V = \dim W + \dim W^\perp.\)
\end{enumerate}
\end{thm}

\begin{lem}[Pythagoras’ Theorem]
Let \(V\) be an inner‐product space and let \(u,v\in V\) be orthogonal.  Then
\[
\|u+v\|^2 \;=\;\|u\|^2 + \|v\|^2.
\]
\end{lem}

\begin{thm}[Projection is Closest Vector in $W$]
Let \(V\) be an inner‐product space, \(W\le V\) finite‐dimensional, and \(u\in V\).  Write the unique decomposition
\[
u = w + w',\quad w\in W,\;w'\in W^\perp,
\]
so that \(w = P_W(u)\).  Then \(w\) is the closest vector in \(W\) to \(u\), in the sense that for every \(z\in W\),
\[
\|u - w\|\;\le\;\|u - z\|.
\]
\end{thm}

\begin{thm}[Gram–Schmidt Orthogonalization]
With the notation of the Gram–Schmidt process applied to a basis \(\{u_1,\dots,u_n\}\subset V\), one defines vectors \(v_1,\dots,v_n\).  Then:
\begin{enumerate}
  \item For each \(k=1,\dots,n\),
  \[
    \mathrm{span}\{v_1,\dots,v_k\}
    =\,\mathrm{span}\{u_1,\dots,u_k\},
  \]
  hence \(\{v_1,\dots,v_n\}\) spans the same subspace \(W\).
  \item For each \(k\), the set \(\{v_1,\dots,v_k\}\) is orthogonal in 
  \(W_k=\mathrm{span}\{u_1,\dots,u_k\}\), and in particular 
  \(\{v_1,\dots,v_n\}\) is an orthogonal basis of \(W\).
\end{enumerate}
\end{thm}

\subsection{Inner Product Defined by a Matrix}

\begin{defn}[Positive Definite Matrix]
Let \(A\in M_{n\times n}(\mathbb{R})\) be symmetric.  \(A\) is called \emph{positive definite} if
\[
u^T A\,u > 0
\quad\text{for all nonzero }u\in\mathbb{R}^n.
\]
\end{defn}

\begin{prop}
Let \(A\in M_{n\times n}(\mathbb R)\) be symmetric and positive definite.  Then
\[
\langle u,v\rangle_A \;=\; u^T A v
\]
defines an inner product on \(\mathbb R^n\).
\end{prop}

\section{Eigenvalues and Diagonalization}

\subsection{Eigenvalues and Eigenvectors}

\begin{defn}[Eigenvector and Eigenvalue]
Let \(T\colon V\to V\) be a linear operator on a vector space \(V\).  A nonzero vector \(u\in V\) is an \emph{eigenvector} of \(T\) if there exists a scalar \(\lambda\in F\) such that
\[
T(u) \;=\;\lambda\,u.
\]
The scalar \(\lambda\) is called the corresponding \emph{eigenvalue}.
\end{defn}

\begin{defn}[Eigenspace]
If \(\lambda\) is an eigenvalue of \(T\colon V\to V\), the \emph{eigenspace} of \(\lambda\) is the subspace
\[
E_\lambda
=\{\,u\in V \mid T(u)=\lambda\,u\}.
\]
\end{defn}

\begin{defn}[Characteristic Polynomial]
Let \(T\colon V\to V\) be a linear operator on an \(n\)-dimensional vector space \(V\).  The \emph{characteristic polynomial} of \(T\) is
\[
C_T(x)
=\det\bigl([T]_\alpha - x\,I\bigr),
\]
where \(\alpha\) is any basis of \(V\) and \([T]_\alpha\) is the matrix of \(T\) relative to \(\alpha\).
\end{defn}

\begin{prop}[Characteristic Polynomial]
Let \(T:V\to V\) be a linear operator on an \(n\)-dimensional space \(V\).  Its characteristic polynomial \(C_T(x)\) satisfies:
\begin{enumerate}
  \item \(C_T(x)\) is independent of the choice of basis.
  \item \(\deg C_T = n\).
  \item A scalar \(\lambda\) is an eigenvalue of \(T\) if and only if \(C_T(\lambda)=0\).
  \item The eigenspace \(E_\lambda = \ker(T-\lambda I)\) is a subspace of \(V\).
\end{enumerate}
\end{prop}

\subsection{Diagonalization}

\begin{defn}[Diagonalizable Operator]
Let \(T\colon V\to V\) be a linear operator on a finite-dimensional vector space \(V\).  We say \(T\) is \emph{diagonalizable} if there exists a basis \(\alpha\) of \(V\) consisting entirely of eigenvectors of \(T\); equivalently, in that basis the matrix of \(T\) is diagonal.
\end{defn}

\begin{prop}[Diagonalizability Criterion]
Let \(T:V\to V\) be a linear operator on a finite‐dimensional space \(V\).  Then \(T\) is diagonalizable if and only if there exists a basis of \(V\) consisting of eigenvectors of \(T\); equivalently, for any basis \(\beta\), the matrix \([T]_\beta\) is similar to a diagonal matrix.
\end{prop}

\begin{defn}[Diagonalizable Matrix]
Let \(A\in M_{n\times n}(F)\).  We say \(A\) is \emph{diagonalizable} if there exist an invertible matrix \(Q\in M_{n\times n}(F)\) and a diagonal matrix \(D\in M_{n\times n}(F)\) such that
\[
Q^{-1}\,A\,Q \;=\; D.
\]
\end{defn}

\subsection{Diagonalizability}
\begin{prop}
Let \(V\) be a finite‐dimensional real vector space and \(T:V\to V\) linear.  If the characteristic polynomial \(C_T(x)\) has a nonreal complex root, then \(T\) is not diagonalizable over \(\mathbb R\).
\end{prop}

\begin{lem}
Let \(T:V\to V\) be linear and let \(\lambda_1\neq\lambda_2\) be two distinct eigenvalues.  Then
\[
E_{\lambda_1}\;\cap\;E_{\lambda_2} \;=\;\{0\}.
\]
\end{lem}

\begin{prop}
Let \(T:V\to V\) have distinct eigenvalues \(\lambda_1,\dots,\lambda_m\).  For each \(i\) let \(\beta_i\) be a basis of the eigenspace \(E_{\lambda_i}\), and set \(\beta=\beta_1\cup\cdots\cup\beta_m\).  Then:
\begin{enumerate}
  \item \(\lvert\beta\rvert = \sum_i \lvert\beta_i\rvert.\)
  \item \(\beta\) is linearly independent.
\end{enumerate}
\end{prop}

\begin{cor}
If \(T:V\to V\) is linear on an \(n\)-dimensional space and has \(n\) distinct eigenvalues, then \(T\) is diagonalizable.
\end{cor}

\begin{defn}[Geometric and Algebraic Multiplicity]
Let \(T\colon V\to V\) be a linear operator and \(\lambda\) an eigenvalue of \(T\).  
\begin{itemize}
  \item The \emph{geometric multiplicity} of \(\lambda\) is \(\dim E_\lambda\), where \(E_\lambda=\ker(T-\lambda I)\).
  \item The \emph{algebraic multiplicity} of \(\lambda\) is the exponent of the factor \((x-\lambda)\) in the characteristic polynomial \(C_T(x)\).
\end{itemize}
\end{defn}

\begin{thm}
Let \(T:V\to V\) be a linear operator on a finite‐dimensional space \(V\), and let \(\lambda\) be any eigenvalue of \(T\).  Then
\[
1 \;\le\;\bigl(\text{geometric multiplicity of }\lambda\bigr)
\;\le\;\bigl(\text{algebraic multiplicity of }\lambda\bigr).
\]
\end{thm}

\begin{prop}
Let \(T:V\to V\) be a linear operator on an \(n\)-dimensional vector space over a field \(F\), with characteristic polynomial \(C_T(x)\).
\begin{enumerate}
  \item If \(F=\mathbb{R}\) and \(C_T(x)\) has a nonreal root, then \(T\) is not diagonalizable over \(\mathbb{R}\).
  \item If \(F=\mathbb{R}\) and all roots of \(C_T(x)\) lie in \(\mathbb{R}\), or if \(F=\mathbb{C}\), then
  \[
  T \text{ is diagonalizable}
  \quad\Longleftrightarrow\quad
  \forall\,\lambda,\;
    \bigl(\text{geom.~mult.~}\lambda\bigr)
    =\bigl(\text{alg.~mult.~}\lambda\bigr).
  \]
  \item When \(T\) is diagonalizable, let \(\lambda_1,\dots,\lambda_k\) be its distinct eigenvalues and \(\beta_i\) a basis of the eigenspace \(E_{\lambda_i}\).  Then
  \[
    \beta = \beta_1\;\cup\;\beta_2\;\cup\cdots\cup\;\beta_k
  \]
  is a basis of \(V\) consisting entirely of eigenvectors of \(T\).
\end{enumerate}
\end{prop}

\subsection{Orthogonal Diagonalization}

\begin{defn}[Symmetric, Orthogonal, Self-Adjoint, Unitary Matrices]
Let \(A\in M_{n\times n}(F)\).
\begin{itemize}
  \item If \(F=\mathbb R\), \(A\) is \emph{symmetric} when \(A^T=A\).
  \item If \(F=\mathbb R\), \(A\) is \emph{orthogonal} when its columns form an orthonormal basis of \(\mathbb R^n\) (equivalently \(A^T=A^{-1}\)).
  \item If \(F=\mathbb C\), \(A\) is \emph{self-adjoint} (Hermitian) when \(A^*=A\).
  \item If \(F=\mathbb C\), \(A\) is \emph{unitary} when its columns form an orthonormal basis of \(\mathbb C^n\) (equivalently \(A^*=A^{-1}\)).
\end{itemize}
\end{defn}

\begin{prop}[Orthogonal and Unitary Matrices]\leavevmode\vspace{-20.5pt}
\begin{enumerate}
  \item If \(A\in M_{n}(\mathbb{R})\), then \(A\) is orthogonal \(\Leftrightarrow A^T = A^{-1}\).
  \item If \(A\in M_{n}(\mathbb{C})\), then \(A\) is unitary \(\Leftrightarrow A^* = A^{-1}\).
\end{enumerate}
\end{prop}

\begin{defn}[Orthogonal and Unitary Diagonalizability]
Let \(A\in M_{n\times n}(F)\).
\begin{itemize}
  \item If \(F=\mathbb R\), \(A\) is \emph{orthogonally diagonalizable} if there exists an orthogonal matrix \(Q\) and a diagonal \(D\) with
  \[
    Q^T\,A\,Q = D
    \quad\Longleftrightarrow\quad
    A = Q\,D\,Q^T.
  \]
  \item If \(F=\mathbb C\), \(A\) is \emph{unitarily diagonalizable} if there exists a unitary matrix \(Q\) and a diagonal \(D\) with
  \[
    Q^*\,A\,Q = D
    \quad\Longleftrightarrow\quad
    A = Q\,D\,Q^*.
  \]
\end{itemize}
\end{defn}

\begin{prop}\leavevmode\vspace{-20.5pt}
\begin{enumerate}
  \item If \(A\in M_{n}(\mathbb{R})\) is orthogonally diagonalizable, then \(A\) is symmetric (\(A^T = A\)).
  \item If \(A\in M_{n}(\mathbb{C})\) is unitarily diagonalizable, then \(A\) is normal (\(A^*A = AA^*\)).
\end{enumerate}
\end{prop}

\begin{lem}\leavevmode\vspace{-20.5pt}
\begin{enumerate}
  \item If \(A\in M_{n}(\mathbb{R})\) and \(u,v\in\mathbb{R}^n\), then
  \[
    \langle A u, v\rangle = \langle u, A^T v\rangle,
    \quad
    \langle u, A v\rangle = \langle A^T u, v\rangle.
  \]
  \item If \(A\in M_{n}(\mathbb{C})\) and \(u,v\in\mathbb{C}^n\), then
  \[
    \langle A u, v\rangle = \langle u, A^* v\rangle,
    \quad
    \langle u, A v\rangle = \langle A^* u, v\rangle.
  \]
\end{enumerate}
\end{lem}

\begin{thm}[Reality of Eigenvalues]
If \(A\in M_n(\mathbb{R})\) is symmetric or \(A\in M_n(\mathbb{C})\) is self-adjoint, then all roots of its characteristic polynomial lie in \(\mathbb{R}\); equivalently, every eigenvalue of \(A\) is real.
\end{thm}

\begin{thm}[Orthogonality of Eigenspaces]
Let \(A\in M_n(\mathbb{R})\) be symmetric or \(A\in M_n(\mathbb{C})\) be normal.  Then its eigenspaces are mutually orthogonal: if \(u\in E_{\lambda_1}\) and \(w\in E_{\lambda_2}\) with \(\lambda_1\neq\lambda_2\), then \(\langle u,w\rangle=0\).
\end{thm}

\begin{thm}[Spectral Theorem]\leavevmode\vspace{-20.5pt}
\begin{enumerate}
  \item For \(A\in M_n(\mathbb{R})\):
  \[
    A \text{ orthogonally diagonalizable}
    \quad\Longleftrightarrow\quad
    A \text{ symmetric }(A^T=A).
  \]
  \item For \(A\in M_n(\mathbb{C})\):
  \[
    A \text{ unitarily diagonalizable}
    \quad\Longleftrightarrow\quad
    A \text{ normal }(A^*A=AA^*).
  \]
\end{enumerate}
\end{thm}

\begin{defn}[Definiteness of a Symmetric Matrix]
Let \(A\in M_{n\times n}(\mathbb{R})\) be symmetric.  We say:
\begin{itemize}
  \item \(A\) is \emph{positive definite} if \(u^T A\,u>0\) for all nonzero \(u\in\mathbb{R}^n\).
  \item \(A\) is \emph{positive semidefinite} if \(u^T A\,u\ge0\) for all \(u\in\mathbb{R}^n\).
  \item \(A\) is \emph{negative definite} if \(u^T A\,u<0\) for all nonzero \(u\).
  \item \(A\) is \emph{negative semidefinite} if \(u^T A\,u\le0\) for all \(u\).
  \item Otherwise, \(A\) is called \emph{indefinite}.
\end{itemize}
\end{defn}

\begin{thm}[Positive Definiteness]
Let \(A\in M_n(\mathbb{R})\) be symmetric.  Then
\[
A \text{ is positive definite}
\quad\Longleftrightarrow\quad
\text{all eigenvalues of }A\text{ are positive}.
\]
\end{thm}

\begin{lem}
Let \(A\in M_{m\times n}(\mathbb{R})\).  Then:
\begin{enumerate}
  \item \(A^T A\in M_{n\times n}(\mathbb{R})\) and \(A A^T\in M_{m\times m}(\mathbb{R})\) are symmetric and positive semi-definite.
  \item \(\displaystyle \mathrm{rank}(A^T A) = \mathrm{rank}(A) = \mathrm{rank}(A A^T).\)
\end{enumerate}
\end{lem}

\subsection{Singular Value Decomposition}

\begin{thm}[Singular Value Decomposition]
Let \(A\in M_{m\times n}(\mathbb{R})\), and let \(L_A:\mathbb{R}^n\to\mathbb{R}^m\) be the associated linear map.  Set \(r=\mathrm{rank}(A)\).  Then there exist orthonormal bases
\[
\beta = \{v_1,\dots,v_n\}\subset\mathbb{R}^n,
\qquad
\gamma=\{w_1,\dots,w_m\}\subset\mathbb{R}^m,
\]
and positive scalars
\(\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_r>0\),
such that
\[
L_A(v_i) =
\begin{cases}
\sigma_i\,w_i, & i=1,\dots,r,\\
0, & i=r+1,\dots,n.
\end{cases}
\]
Consequently, the matrix of \(L_A\) relative to \(\beta,\gamma\) is
\[
[L_A]_\beta^\gamma
=
\begin{pmatrix}
\sigma_1 &        &        &        &        &        \\
         & \sigma_2 &        &        &        &        \\
         &        & \ddots &        &        &        \\
         &        &        & \sigma_r &        &        \\
         &        &        &        & 0      &        \\
         &        &        &        &        & \ddots 
\end{pmatrix}_{m\times n}.
\]
\end{thm}

\begin{defn}[Singular Values]
Let \(A\in M_{m\times n}(\mathbb{R})\) have rank \(r\).  In its singular value decomposition one obtains positive scalars
\[
\sigma_1 \;\ge\;\sigma_2\;\ge\;\cdots\;\ge\;\sigma_r>0.
\]
These \(\sigma_1,\dots,\sigma_r\) are called the \emph{singular values} of \(A\).
\end{defn}

\begin{cor}
In the SVD of Theorem 117:
\begin{enumerate}
  \item \(\{v_1,\dots,v_n\}\) is an orthonormal basis of eigenvectors of \(A^T A\) in \(\mathbb{R}^n\).
  \item \(\{w_1,\dots,w_m\}\) is an orthonormal basis of eigenvectors of \(A A^T\) in \(\mathbb{R}^m\).
  \item \(A^T A\) and \(A A^T\) have the same nonzero eigenvalues, and the singular values \(\sigma_i\) are the positive square roots of these eigenvalues.
\end{enumerate}
\end{cor}

\clearpage

% === License page ===
\phantomsection
\addcontentsline{toc}{section}{License}
\begin{center}
  \Huge \textbf{License}
\end{center}

\vspace{2em}

\noindent\textbf{MIT License}

\vspace{1em}

\noindent Copyright (c) 2025 Cagri Arslan

\vspace{1em}

\noindent Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

\vspace{1em}

\noindent The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

\vspace{1em}

\noindent THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.



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