Matrix multiplication as composition

What to learn

  • How to do Matrix multiplication

Description

  • We already know multipling matrix by vector.
  • Multiple transformation using matrixs can be interpreted by Only one matrix.

  • Think about composite function like f(g(x)). we read this from right to left. matrix multiplication has same property. Make imaginary basis(i hat and j hat) and multipling matrix by vector from right to left.

    \[\begin{bmatrix}a & b\\ c & d\end{bmatrix}\begin{bmatrix}e & f\\ g & h\end{bmatrix} = \begin{bmatrix}a & b\\ c & d\end{bmatrix}\begin{bmatrix}e\\ g\end{bmatrix} , \begin{bmatrix}a & b\\ c & d\end{bmatrix}\begin{bmatrix}f\\ h\end{bmatrix}\] \[\begin{bmatrix}a & b\\ c & d\end{bmatrix}\begin{bmatrix}e\\ g\end{bmatrix} , \begin{bmatrix}a & b\\ c & d\end{bmatrix}\begin{bmatrix}f\\ h\end{bmatrix} = e \begin{bmatrix}a\\ b\end{bmatrix} + g \begin{bmatrix}b\\ d\end{bmatrix} , f \begin{bmatrix}a\\ b\end{bmatrix} + h \begin{bmatrix}b\\ d\end{bmatrix}\] \[e \begin{bmatrix}a\\ c\end{bmatrix} + g \begin{bmatrix}b\\ d\end{bmatrix} , f \begin{bmatrix}a\\ c\end{bmatrix} + h \begin{bmatrix}b\\ d\end{bmatrix} = \begin{bmatrix} ae+bg & af+bh\\ ce+dg & cf+dh \end{bmatrix}\]

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