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% Written By <NAME>
% Class: Linear Algebra (MATH-194)
% Professor: <NAME>
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\title{Matrices for Linear Transformations}
\date{\today}
\author{<NAME>\\ \small Professor: <NAME>}

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\begin{document}

\maketitle

\begin{itemize}

  \item Standard Matrix for linear transformations:

    \begin{enumerate}

      \item ex. $T:\,\mathbb{R}^2\rightarrow\mathbb{R}^3$. $T\left( \begin{bmatrix} x_1\\x_2\end{bmatrix} \right)=\begin{bmatrix} x_1+2x_2\\3x_1+4x_2\\5x_1+6x_2\end{bmatrix}$, then $T(\overrightarrow{x})=\begin{bmatrix} 1 & 2\\3&4\\5&6\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=A\overrightarrow{x}$, where $A=\begin{bmatrix}1&2\\3&4\\5&6\end{bmatrix}$

      \item Standard Matrix for $T:\,\mathbb{R}^n\rightarrow\mathbb{R}^m$, where $T$ is a linear transformation and $B=\left\{ \overrightarrow{e}_1,\overrightarrow{e}_2,\dots,\overrightarrow{e}_n \right\}$ is a standard basis for $\mathbb{R}^n$, then $A=\left[ T(\overrightarrow{e}_1)\,\Big|\,T(\overrightarrow{e}_2)\,\Big|\,\dots\,\Big|\,T(\overrightarrow{e}_n) \right]$ is a standard matrix for $T$ such that $T(\overrightarrow{x})=A\overrightarrow{x} \,\,\,\forall\overrightarrow{x}\in\mathbb{R}^n$

      \item Note: Every linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ is a matrix transformation

    \end{enumerate}
    
  \item Let $V$ and $W$ be vector spaces with bases $B$ and $B'$, where $B=\left\{ \overrightarrow{v}_1,\overrightarrow{v}_2,\dots,\overrightarrow{v}_n \right\}$. If $T:\,V\rightarrow W$ is a linear transformation such that:\newline $[T(\overrightarrow{v}_1)]_{B'}=\begin{bmatrix} a_{11}\\a_{21}\\\vdots\\a_{31}\end{bmatrix},[T(\overrightarrow{v}_2)]_{B'}=\begin{bmatrix} a_{12}\\a_{22}\\\vdots\\a_{32}\end{bmatrix},\dots,[T(\overrightarrow{v}_B)]_{B'}=\begin{bmatrix}a_{1n}\\a_{2n}\\\vdots\\a_{mn}\end{bmatrix}$. Then, the $m\times n$ matrix whose $n$ columns correspond to $[T(\overrightarrow{v}_i)]_{B'}$ is such that $[T(\overrightarrow{v})]_{B'}=A[\overrightarrow{v}]_B$

\end{itemize}

\end{document}

