1.11 Transformation Matrix in New Basis

Lets look at how transformation matrices of a linear mapping $T: V \to W$ change if we change the bases in $V$ and W.

Consider two ordered bases of  $V$

$B= ( b_1,b_2,......,b_n)$

$\tilde{B}=( \tilde{b_1},\tilde{b_2},......,\tilde{b_n})$

Consider two ordered bases of W

$C= ( c_1,c_2,......,c_n)$

$\tilde{C}=( \tilde{c_1},\tilde{c_2},......,\tilde{c_m})$

Let $A\in \mathbb{R}^{m \times n}$ be the transformation matrix from $V \to W $ with respect to basic $B$ and $C$ and $\tilde{A}\in \mathbb{R}^{m \times n}$ be the transformation matrix from $V \to W$ with respect to basis $\tilde{B}$ and $\tilde{C}$.

We will find out how $A$ and $\tilde{A}$ are related ie; how we can transform $A$ to $\tilde{A}$.

The transformation matrix $\tilde{A}=T^{-1} A S $

Here $S \in \mathbb{R}^{n \times n}$ is the  transformation matrix that maps coordinates with respect to $\tilde{B}$ into coordinates with respect to $B$ and  $T \in \mathbb{R}^{m \times m}$ is the  transformation matrix that maps coordinates with respect to $\tilde{C}$ into coordinates with respect to $C$.

Lets consider a transformation matrix with respect to the standard basis(B)

$A=\displaystyle \left[\begin{matrix}1 & 2\\2 & 1\end{matrix}\right]$

If we define a new basis  in $\mathbb{R}^2$

$S=\displaystyle \left[\begin{matrix}1 & 1\\1 & -1\end{matrix}\right]$

 

Then any vector in the new basis can be transformed with a new transformation matrix which is easier to work $\tilde{A}=S^{-1}AS$

     $\tilde{A}=  \left[\begin{matrix}-1 & -1\\-1 & 1\end{matrix}\right]  \left[\begin{matrix}1 & 2\\2 & 1\end{matrix}\right]  \left[\begin{matrix}1 & 1\\1 & -1\end{matrix}\right] =  \left[\begin{matrix}3 & 0\\0 & 1\end{matrix}\right]$

It is noted that we obtain a diagonal matrix which is easier to work than the original transformation matrix.

import numpy as np

B=np.array([[1,1],[1,-1]])

A=np.array([[2,1],[1,2]])

Ab=np.linalg.inv(B).dot(A.dot(B))

print("new transformation matrix in base B")

print(Ab)

o/p

new transformation matrix in base B 

[[3. 0.] 

[0. 1.]]

import numpy as np

Bn=np.array([[1,0,1],[1,1,0],[0,1,1]])

Cn=np.array([[1,1,0,1],[1,0,1,0],[0,1,1,0],[0,0,0,1]])

A=np.array([[1,2,0],[-1,1,3],[3,7,1],[-1,2,4]])

print("Tranformation Matrix A defined in stardard basis")

print(A)

S=Bn

T=Cn

print("New Basis")

print(S)

print(T)

print("Tranformation matrix in New basis is")

An=A.dot(S)

An=np.linalg.inv(T).dot(An)

print(An)

o/p

Transformation Matrix A defined in standard basis
[[ 1  2  0]
 [-1  1  3]
 [ 3  7  1]
 [-1  2  4]]
New Basis-S

[[1 0 1]
 [1 1 0]
 [0 1 1]]

New Basis-T
[[1 1 0 1]
 [1 0 1 0]
 [0 1 1 0]
 [0 0 0 1]]
Transformation matrix in New basis is
[[-4. -4. -2.]
 [ 6.  0.  0.]
 [ 4.  8.  4.]
 [ 1.  6.  3.]]

Example:

Given the linear transformation $T(x,y,z,w)=(x+2y-z,x+w)$.Find the matrix representation of the above transformation. Find the matrix representation with respect to the bases $B_1=\{(1,0,0,0),(1,1,0,0),(1,1,1,0),(1,1,1,1)\}$ for $\mathbb{R}^4$ and $B_2=\{(1,1),(1,0)\}$ for $\mathbb{R}^2$

Transformation matrix in standard basis

$A=\begin{bmatrix}
1&2&-1&0 \\
1&0&0&1
\end{bmatrix}$

New Basis transformation matrix

$S=\begin{bmatrix}
1&1&1&1 \\
0&1&1&1\\
0&0&1&1\\
0&0&0&1
\end{bmatrix}$

New Basis transformation matrix

$T=\begin{bmatrix}
1&1 \\
1&0\\
\end{bmatrix}$
Inverse of $T$ is
$T=\begin{bmatrix}
0&1 \\
1&-1\\
\end{bmatrix}$

So the transformation matrix in new basis is  $\tilde{A}=T^{-1} A S $

$\tilde{A}=\begin{bmatrix}0&1 \\
1&-1\\
\end{bmatrix} \begin{bmatrix}
1&2&-1&0 \\
1&0&0&1
\end{bmatrix}\begin{bmatrix}
1&1&1&1 \\
0&1&1&1\\
0&0&1&1\\
0&0&0&1
\end{bmatrix}$

$\tilde{A}=\begin{bmatrix}
1&1&1&2 \\
0&2&1&0
\end{bmatrix}$