Find Matrix of Linear Transformation T w/ Respect to Basis B

Once you have P, you can find B=P^{-1}AP, which will give you the matrix of T with respect to basis B. In summary, the matrix of T with respect to the basis B is -28 -19 -43, 5 4 7, 18 11 28.
  • #1
kiwifruit
8
0
I came across this problem in one of my linear algebra books.
A linear transformation T:R^3 ->R^3 has matrix

2 3 0
-1 1 2
2 0 1
with respect to the standard basis for R^3. Find the matrix of T with respect to the basis
B={(1,2,1),(0,1,-1),(2,3,2)}

The answer given is
-28 -19 -43
5 4 7
18 11 28
but i have no idea how to get to that answer as the book does not provide workings/steps. Any help would be appreciated thanks.
 
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  • #2
I recommend that you read this post (the part above the quote) to make sure that you understand the relationship between linear operators and matrices.
 
  • #3
I'll use subscript B to indicate a vector or linear transformation expressed as a matrix in the new basis, thus

[tex]\left ( Tx \right )_B = T_B x_B[/tex]

Let B be a matrix whose columns are the basis vectors of the new basis, expressed in the standard basis. Then the inverse of B will convert the components of a general vector from the standard basis to the new basis:

[tex]B^{-1}Tx = T_B B^{-1}x.[/tex]

So [itex]B^{-1}T[/itex] has the same effect on [itex]x[/itex] as [itex]T_B B^{-1}[/itex]. Now all we have to do is solve for [itex]T_B[/itex].

[tex]B^{-1}T = T_B B^{-1}[/itex]

[tex]B^{-1}TB = T_B.[/itex]
 
  • #4
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  • #5
Say vectors [itex]\vec{x}[/itex] and [itex]\vec{y}[/itex] have the representations

[tex]
\vec{x}=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}_1=\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2
\hspace{0.5in}
\vec{y}=\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}_1=\begin{pmatrix}y_1'\\y_2'\\y_3'\end{pmatrix}_2[/tex]

with respect to basis 1 and basis 2. You can construct a matrix P that will convert between the two representations:

[tex]\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}_1=P\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2[/tex]

and its inverse P-1 will take you in the other direction:

[tex]\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2=P^{-1}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}_1[/tex]

If [itex]\vec{y}=T(\vec{x})[/itex], there are matrices A and B such that

[tex]\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}_1=A\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}_1
\hspace{0.5in}
\begin{pmatrix}y_1'\\y_2'\\y_3'\end{pmatrix}_2=B\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2
[/tex].

It turns out that A and B are related by [itex]B=P^{-1}AP[/itex] because

[tex]\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}_1=P\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2[/tex]

[tex]\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}_1=A\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}_1=AP\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2[/tex]

[tex]\begin{pmatrix}y_1'\\y_2'\\y_3'\end{pmatrix}_2=P^{-1}\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}_1=P^{-1}AP\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2[/tex]

In your problem, you're given A, and you want to find B. So the problem boils down to finding P given the information you have about the two bases.
 

What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the underlying structure of the vector space.

What is a matrix representation of a linear transformation?

A matrix representation of a linear transformation is a way to represent the transformation using a matrix. Each column of the matrix represents the image of the basis vectors under the transformation.

What is a basis?

A basis is a set of linearly independent vectors that span a vector space. It is used to represent other vectors in the space and can be used to find the matrix representation of a linear transformation.

How do you find the matrix of a linear transformation with respect to a given basis?

To find the matrix of a linear transformation with respect to a given basis, you first need to find the images of the basis vectors under the transformation. Then, you can arrange these images as columns in a matrix to get the matrix representation of the transformation.

Why is finding the matrix of a linear transformation important?

Finding the matrix of a linear transformation allows us to easily perform calculations and analyze the properties of the transformation. It also allows us to compare and contrast different transformations and determine if they have similar properties.

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