[Linear Algebra] 12. Change of basis

Standard way to describe vector is using coordinates $\left[\begin{array}{c}x\\ y\end{array}\right]$. each coordinate is scalar to stretches or squishes basis vectors, normally $\hat{\mathbf{i}}$ and $\hat{\mathbf{j}}$ which are standard basis vectors.

 

In this section, think about the idea of using different set of basis vectors.

 

Let's define new basis vectors $\overrightarrow{{\mathbf{b}}_{\mathbf{1}}}$ and $\overrightarrow{{\mathbf{b}}_{\mathbf{2}}}$ which are $\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]$ and $\left[\begin{array}{c}{w}_1\\ w_2\end{array}\right]$ described by standard perspective. But in new perspective, those vectors have coordinates $\left[\begin{array}{c}1\\ 0\end{array}\right]$ and $\left[\begin{array}{c}0\\ 1\end{array}\right]$.

 

Even though we are looking at same vectors, we use different vectors to describe them. Basis vectors are nothing more than tool to describe vectors.

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Translate between coordinates systems

new language $\to$ standard language
If new perspective describes a vector with coordinates $\left[\begin{array}{c}x\\ y\end{array}\right]$, this vector is $x\overrightarrow{{\mathbf{b}}_{\mathbf{1}}}+y\overrightarrow{{\mathbf{b}}_{\mathbf{2}}}$. Because $\overrightarrow{{\mathbf{b}}_{\mathbf{1}}}$ is $\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]$ and $\overrightarrow{{\mathbf{b}}_{\mathbf{2}}}$ is $\left[\begin{array}{c}{w}_1\\ w_2\end{array}\right]$ in standard perspective, this vector is $(x{v}_1+y{w}_1)\hat{\mathbf{i}}+(x{v}_2+y{w}_2)\hat{\mathbf{j}}$. This process is matrix-vector multiplication:

$$\left[\begin{array}{cc}{v}_1& w_1\\ v_2& w_2\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$

Geometrically, this matrix translates standard grid into new grid. $⟷$ Numerically, it translates a vector in new language to standard language.

 

standard language $\to$ new language
In this case, use inverse matrix:

$$\left[\begin{array}{cc}{v}_1& w_1\\ v_2& w_2\end{array}\right]\left[\begin{array}{c}{x}_{new}\\ y_{new}\end{array}\right]=\left[\begin{array}{c}{x}_{standard}\\ y_{standard}\end{array}\right]⟷\left[\begin{array}{c}{x}_{new}\\ y_{new}\end{array}\right]={\left[\begin{array}{cc}{v}_1& w_1\\ v_2& w_2\end{array}\right]}^{-1}\left[\begin{array}{c}{x}_{standard}\\ y_{standard}\end{array}\right]$$

 

This is a way to translate the description of individual vectors back and forth between different languages.

Representing same transformation in both standard and new language

For example, $\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$ represents ${90}^{\circ }$ rotation of space. It means that where $\hat{\mathbf{i}}$ and $\hat{\mathbf{j}}$ land, not $\overrightarrow{{\mathbf{b}}_{\mathbf{1}}}$ and $\overrightarrow{{\mathbf{b}}_{\mathbf{2}}}$ land.

 

Common way to represent same transformation in new language: 0). start with vector written in new language. 1). translate it into standard language, 2). apply the transformation matrix, 3). apply inverse change of basis matrix to get into new language:

$${\left[\begin{array}{cc}{v}_1& w_1\\ v_2& w_2\end{array}\right]}^{-1}\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\left[\begin{array}{cc}{v}_1& w_1\\ v_2& w_2\end{array}\right]\left[\begin{array}{c}{x}_{new}\\ y_{new}\end{array}\right]$$

 

In general, expression like $V^{-1}AV$ suggests a mathematical empathy. 1). $A$ represents some transformation, and 2). $V^{-1}$ and $V$ represent the shift of perspective, And 3). $V^{-1}AV$ represents that same transformation in different language.