Main Concept

A linear transformation on a vector space is an operation $T$ on the vector space satisfying two rules:

$T\left(\overrightarrow{\mathit{x}}\+\overrightarrow{\mathit{y}}\right)\=T\left(\overrightarrow{\mathit{x}}\right)\+T\left(\overrightarrow{\mathit{y}}\right)$,

$T\left(\mathrm{\α} \overrightarrow{\mathit{x}}\right)\=\mathrm{\α} T\left(\overrightarrow{\mathit{x}}\right)$

for all vectors $\overrightarrow{\mathit{x}}$, $\overrightarrow{\mathit{y}}$, and all scalars $\mathrm{\α}$.

Any linear transformation $T$ in the Euclidean plane is characterized by the action of that transformation on the standard basis:

$T\left(\overrightarrow{\mathit{x}}\right) \= T\left(x_1\stackrel{\mathbf{\wedge }}{\mathit{i}}\+x_{2 }\stackrel{\mathbf{\wedge }}{\mathit{j}}\right)\=x_{1}T\left(\stackrel{\mathbf{\wedge }}{\mathit{i}}\right)\+x_{2}T\left(\stackrel{\mathbf{\wedge }}{\mathit{j}}\right)$

$\=\mathit{A} \. \overrightarrow{\mathit{x}}$

where

$\mathit{A}\=\left[\begin{array}{cc}T\left(\stackrel{\mathbf{\wedge }}{\mathit{i}}\right)& T\left(\stackrel{\mathbf{\wedge }}{\mathit{j}}\right)\end{array}\right]$,   $\overrightarrow{\mathit{x}} \= \left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$,   $\stackrel{\mathbf{\wedge }}{\mathit{i}}\mathbf{\=}\left[\begin{array}{c}1\\ 0\end{array}\right]$,    $\stackrel{\mathbf{\wedge }}{\mathit{j}}\=\left[\begin{array}{c}0\\ 1\end{array}\right]$.

The matrix $\mathit{A}$, whose columns are the transformed basis vectors, is known as the transformation matrix associated to the transformation $T$.

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