What is a linear transformation?
\[
\huge{T: V \rightarrow W}
\]
\[
\huge{T: \mathbb{R}^{n} \rightarrow \mathbb{R}^m}
\]
\[
\huge{T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}}
\]
\[
\beta = \left \{ \begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}0\\1\end{bmatrix} \right\}
\]
\[
\beta = \left \{ \begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}0\\1\end{bmatrix} \right\}
\]
\[
[T]_{\beta}^{\beta} = \begin{bmatrix}
2 & 0\\
0 & 2
\end{bmatrix}
\]
\[
\beta = \left \{ \begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}0\\1\end{bmatrix} \right\}
\]
\[
[T]_{\beta}^{\beta} = \begin{bmatrix}
0 & -1\\
1 & 0
\end{bmatrix}
\]
\[
\beta = \left \{ \begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}0\\1\end{bmatrix} \right\}
\]
\[
[T]_{\beta}^{\beta} = \begin{bmatrix}
0 & -2\\
2 & 0
\end{bmatrix}
\]
What is the linear transformation that we just saw?
What is the linear transformation that we just saw?
It is a composition of two transformations. What are they?
What is the linear transformation that we just saw?
It is a composition of two transformations. What are they?
\[
T_1 = \begin{bmatrix}
2 & 0\\
0 & 2
\end{bmatrix}, T_2 = \begin{bmatrix}
0 & -1\\
1 & 0
\end{bmatrix}
\]
What is the linear transformation that we just saw?
It is a composition of two transformations. What are they?
\[
T_1 = \begin{bmatrix}
2 & 0\\
0 & 2
\end{bmatrix}, T_2 = \begin{bmatrix}
0 & -1\\
1 & 0
\end{bmatrix}
\]
\[
T_1 T_2 = \begin{bmatrix}
2 & 0\\
0 & 2
\end{bmatrix} \begin{bmatrix}
0 & -1\\
1 & 0
\end{bmatrix} = \begin{bmatrix}
0 & -2\\
2 & 0
\end{bmatrix}
\]
\[
\beta = \left \{ \begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}0\\1\end{bmatrix} \right\}
\]
\[
[T]_{\beta}^{\beta} = \begin{bmatrix}
2 & 1\\
1 & 2
\end{bmatrix}
\]