Matrices

Matrix Multiplication

• Composition of a rotation and a shear

Shear times (Rotation times vector) The composition is the same net result of both operations.

Like: \(f(g(x))\)

\[ M_{2} : \begin{bmatrix} 0 & 2 \\ 1 & 0 \end{bmatrix} \cdot M_{1}: \begin{bmatrix} 1 & -2 \\ 1 & 0 \end{bmatrix}\]

\[\begin{bmatrix} 0 & 2 \\ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix} \]

\[\begin{bmatrix} 0 & 2 \\ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} -2 \\ 0 \end{bmatrix} = -2 \begin{bmatrix} 0 \\ 1 \end{bmatrix} + 0 \begin{bmatrix} 2 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ -2 \end{bmatrix} \]

\[ M_{2} \cdot M_{1} = \begin{bmatrix} 2 & 0 \\ 1 & -2 \end{bmatrix}\]

1. Where does \(\hat{i}\) go?
2. The first column of \(M_{1}\) is \(\hat{i}\).
3. The first column of the composition matrix will be the result of \(M_{2} \cdot \hat{i}\).
4. Now, where does \(\hat{j}\) go?
5. The second column of \(M_{1}\) is \(\hat{j}\).
6. The second column of the composition matrix will be the result of \(M_{2} \cdot \hat{j}\).

Matrix multiplication is about transformations.

The order of transformations matters!

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