In the previous lesson, we learned about how vectors are linearly transformed using matrices. This section deals with transformations of transformations.
Let us start with vector $V = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$ transformed by the matrix $ A = \begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix}$ and then by $ B = \begin{bmatrix} 0 & 2 \\ -1 & 3 \end{bmatrix}$; that is, we apply both transformations $A$ and $B$ to $V$ successively. Through the methods of the previous section, it follows that, $ \begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix}\begin{bmatrix} 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 7 \\ 2 \end{bmatrix} $ and $ \begin{bmatrix} 0 & 2 \\ -1 & 3 \end{bmatrix}\begin{bmatrix} 7 \\ 2 \end{bmatrix} = \begin{bmatrix} 4 \\ -1 \end{bmatrix}$. Thus, applying both transformations $A$ and $B$ moves vector $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$ to $\begin{bmatrix} 4 \\ -1 \end{bmatrix}$. There is also an another matrix
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