Linear Algebra — Matrices and Linear Transformations

🛠 Tools and Programming
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笔记

线性变换（linear transformation）本质就是一种function————只不过是接受一个vector input and spit out another vector

既然本质一样，为什么要专门发明linear transformation这个词？the word "trasformation"suggests that you think using movement

如果某种transformation拥有以下两种feature，就称其为linear transformation

  • lines remains lines
  • origin remains fixed

how to describe these transformations numerically？Acctually，you only need to record two basis vectors（ \hat{\imath} & $\hat{\jmath}$），then everything else will follow from that. We can package these coordinates into a 2 \times 2 matrix, such as

\left[ \begin{pmatrix} 3 \\ -2 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \end{pmatrix} \right] = \begin{bmatrix} 3 & 2 \\ -2 & 1 \end{bmatrix}

如果对一个向量（x，y）进行线性变换（矩阵就是变换后的基向量），最终会获得一个变换后的向量，这个过程在数学表达上相当于进行了一次矩阵乘法

\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = x \begin{bmatrix} a \\ c \end{bmatrix} + y \begin{bmatrix} b \\ d \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}

here is some examples

  • 90度逆时针旋转 90° rotation counterclockwise
\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}
  • 剪切 shear
\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}

To sum up， linear transformation are way to move around space such the grid lines remain parallel and evenly spaced and such that the origin remains fixed。Every time you see a linear transformation matrix， you can interpret it as a certain transformation of space。

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