\subsection{Homogeneous Linear System}

\begin{defn}[Homogeneous Linear Equation \& System]\ where 
  \begin{itemize}
    \item Homogeneous Linear Equation:  $a_1x_1 + a_2x_2 + ... + a_nx_n = 0 \iff x_1 = 0, x_2 = 0,... , x_n = 0$
    \item Homogeneous Linear Equation: $\begin{cases}
        a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = 0 \\
        a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = 0 \\ 
        \vdots \\
        a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = 0 \\ 
      \end{cases}$

    \item This is the trivial solution of a homogeneous linear system.
  \end{itemize}

  You can use this to solve problems like Find the equation $ax^2 + by^2 + cz^2 = d$, in the $xyz$ plane which contains the points $(1, 1, -1), (1, 3, 3), (-2, 0, 2)$.

  \begin{itemize}
    \item Solve by first converting to Augmented Matrix, where the last column is all 0. During working steps, this column can be omitted.
    \item With the \hyperref[def:rref]{RREF}, you can set $d$ as $t$ and get values for $a, b, c$ in terms of $t$.
    \item sub in $t$ into the original equation and factorize $t$ out from both sides, for values where $t \neq 0$
  \end{itemize}

\end{defn}

\subsection{Matrix}

\begin{defn}[Matrix]\ \\
  \begin{itemize}
    \item $\begin{pmatrix}
        a_{11} & a_{12} & ... & a_{1n} \\
        a_{21} & a_{22} & ... & a_{2n} \\
        \vdots \\
        a_{m1} & a_{m2} & ... & a_{mn}
      \end{pmatrix}$
    \item $m$ is no of rows, $n$ is no of columns
    \item size is $m \times n$
    \item $A = (a_{ij})_{m \times n}$

  \end{itemize}
\end{defn}

\subsection{Special Matrix}

\begin{note}[Special Matrices]\ \\
  \begin{itemize}
    \item Row Matrix : $\begin{pmatrix} 2 & 1 & 0 \end{pmatrix}$
    \item Column Matrix 
      \subitem $\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}$
    \item \textbf{Square Matrix}, $n \times n$ matrix / matrix of order $n$.
      \subitem Let $A = (a_{ij})$ be a square matrix of order $n$
    \item Diagonal of $A$ is $a_{11}, a_{22}, ..., a_{nn}$.
    \item \textbf{Diagonal Matrix} if Square Matrix and non-diagonal entries are zero
      \subitem Diagonals can be zero
      \subitem \textbf{Identity Matrix} is a special case of this
    \item \textbf{Square Matrix} if Diagonal Matrix and diagonal entries are all the same.
    \item \textbf{Identity Matrix} if Scalar Matrix and diagonal = 1
      \subitem $I_n$ is the identity matrix of order $n$.
    \item \textbf{Zero Matrix} if all entries are 0.
      \subitem Can denote by either $\overrightarrow{0}, 0$
    \item Square matrix is \textbf{symmetric} if symmetric wrt diagonal
      \subitem $A = (a_{ij})_{n \times n}$ is symmetric $\iff a_{ij} = a_{ji},\ \forall i, j$
    \item \textbf{Upper Triangular} if all entries \textbf{below} diagonal are zero.
      \subitem $A = (a_{ij})_{n \times n}$ is upper triangular $\iff a_{ij} = 0 \text{ if } i > j$
    \item \textbf{Lower Triangular} if all entries \textbf{above} diagonal are zero.
      \label{def:ltm}
      \subitem $A = (a_{ij})_{n \times n}$ is lower triangular $\iff a_{ij} = 0 \text{ if } i < j$
      \subitem if Matrix is both Lower and Upper triangular, its a Diagonal Matrix.
  \end{itemize}
\end{note}

\subsection{Matrix Operations}

\begin{defn}[Matrix Operations]\ \\
  Let $A = (a_{ij})_{m \times n}, B = (b_{ij})_{m \times n}$
  \begin{itemize}
    \item Equality: $B = (b_{ij})_{p \times q}$, $A = B \iff m = p \ \& \ n = q \ \& \ a_{ij} = b_{ij} \forall i,j$
    \item Addition: $A + B = (a_{ij} + b_{ij})_{m \times n}$
    \item Subtraction: $A - B = (a_{ij} - b_{ij})_{m \times n}$
    \item Scalar Mult: $cA = (ca_{ij})_{m \times n}$
  \end{itemize}
\end{defn}

\begin{defn}[Matrix Multiplication] \ \\
  Let $A = (a_{ij})_{m \times p}, B = (b_{ij})_{p \times n}$
  \begin{itemize}
    \item $AB$ is the $m \times n$ matrix s.t. $(i,j)$ entry is $$a_{i1}b_{1j} + a_{i2}b_{2j} + ... + a_{ip}b_{pj} = \sum^p_{k=1}a_{ik}b_{kj}$$
    \item No of columns in $A$ = No of rows in $B$.
    \item Matrix multiplication is \textbf{NOT commutative}
  \end{itemize}
\end{defn}