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Transformation Matrix

 Main Concept A linear transformation on a vector space is an operation $T$ on the vector space satisfying two rules:   $T\left(\stackrel{\to }{\mathbit{x}}+\stackrel{\to }{\mathbit{y}}\right)=T\left(\stackrel{\to }{\mathbit{x}}\right)+T\left(\stackrel{\to }{\mathbit{y}}\right)$, for all vectors $\stackrel{\to }{\mathbit{x}}$, $\stackrel{\to }{\mathbit{y}}$, 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:     where $\mathbit{A}=\left[\begin{array}{cc}T\left(\stackrel{\mathbf{\wedge }}{\mathbit{i}}\right)& T\left(\stackrel{\mathbf{\wedge }}{\mathbit{j}}\right)\end{array}\right]$,   ,   $\stackrel{\mathbf{\wedge }}{\mathbit{i}}\mathbf{=}\left[\begin{array}{c}1\\ 0\end{array}\right]$,    $\stackrel{\mathbf{\wedge }}{\mathbit{j}}=\left[\begin{array}{c}0\\ 1\end{array}\right]$.   The matrix $\mathbit{A}$, whose columns are the transformed basis vectors, is known as the transformation matrix associated to the transformation $T$.

Click and/or drag on the graph to change the initial vector $\stackrel{\mathit{\to }}{x}$ or the transformation vectors A$\mathit{\cdot }$$\stackrel{\mathit{\wedge }}{i}$ and A$\mathit{\cdot }$$\stackrel{\mathit{\wedge }}{j}$. You can also edit the values of the transformation matrix A and the vector $\stackrel{\mathit{\to }}{x}$ directly.

 $\mathbit{A}$ $\stackrel{\mathbf{\to }}{\mathbit{x}}$ $\mathbf{=}$ $\mathbit{A}\mathbf{\cdot }\stackrel{\mathbf{\to }}{\mathbit{x}}$  $\mathbf{=}$



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