[Linear Algebra] 12. Change of basis

Standard way to describe vector is using coordinates $\begin{bmatrix} x \\ y \end{bmatrix}$. each coordinate is scalar to stretches or squishes basis vectors, normally $\mathbf{\hat{i}}$ and $\mathbf{\hat{j}}$ 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 $\mathbf{\vec{b_1}}$ and $\mathbf{\vec{b_2}}$ which are $\begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$ and $\begin{bmatrix} w_1 \\ w_2 \end{bmatrix}$ described by standard perspective. But in new perspective, those vectors have coordinates $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$.

 

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 $\rightarrow$ standard language
If new perspective describes a vector with coordinates $\begin{bmatrix} x \\ y \end{bmatrix}$, this vector is $x \mathbf{\vec{b_1}} + y \mathbf{\vec{b_2}}$. Because $\mathbf{\vec{b_1}}$ is $\begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$ and $\mathbf{\vec{b_2}}$ is $\begin{bmatrix} w_1 \\ w_2 \end{bmatrix}$ in standard perspective, this vector is $(xv_1 + yw_1)\mathbf{\hat{i}} + (xv_2 + yw_2)\mathbf{\hat{j}}$. This process is matrix-vector multiplication:

$$
\begin{bmatrix}
v_1 & w_1 \\
v_2 & w_2 \\
\end{bmatrix}
\begin{bmatrix}
x \\ y
\end{bmatrix}
$$

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

 

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

$$
\begin{bmatrix}
v_1 & w_1 \\
v_2 & w_2 \\
\end{bmatrix}
\begin{bmatrix}
x_{new} \\ y_{new}
\end{bmatrix}
=
\begin{bmatrix}
x_{standard} \\ y_{standard}
\end{bmatrix}
\longleftrightarrow
\begin{bmatrix}
x_{new} \\ y_{new}
\end{bmatrix}
=
\begin{bmatrix}
v_1 & w_1 \\
v_2 & w_2 \\
\end{bmatrix}^{-1}
\begin{bmatrix}
x_{standard} \\ y_{standard}
\end{bmatrix}
$$

 

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, $\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ represents $90^{\circ}$ rotation of space. It means that where $\mathbf{\hat{i}}$ and $\mathbf{\hat{j}}$ land, not $\mathbf{\vec{b_1}}$ and $\mathbf{\vec{b_2}}$ 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:

$$
\begin{bmatrix}
v_1 & w_1 \\
v_2 & w_2 \\
\end{bmatrix}^{-1}
\begin{bmatrix}
0 & 1 \\
-1 & 0 \\
\end{bmatrix}
\begin{bmatrix}
v_1 & w_1 \\
v_2 & w_2 \\
\end{bmatrix}
\begin{bmatrix}
x_{new} \\ y_{new}
\end{bmatrix}
$$

 

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.