Matrices of Linear Transformations .... Poole, Example 6.76 ....

In summary, the conversation discusses Example 6.76 from David Poole's book "Linear Algebra: A Modern Introduction" and raises a question about the bases used in calculating the transformation outputs. It is noted that the standard unit vectors are swapped in the non-standard bases \mathcal{B} and \mathcal{C}, which is reflected in the resulting matrix.
  • #1
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I am reading David Poole's book: "Linear Algebra: A Modern Introduction" (Third Edition) and am currently focused on Section 6.6: The Matrix of a Linear Transformation ... ...

I need some help in order to fully understand Example 6.76 ... ...

Example 6.76 reads as follows:View attachment 8770
View attachment 8771
My question or issue of concern is as follows:When we calculate transformation outputs from inputs using \(\displaystyle T \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x - 2y \\ x + y - 3z \end{bmatrix}\)

... it appears, as if by default, that we are using \(\displaystyle \{ e_1, e_2 , e_3 \}\) and \(\displaystyle \{ e_1, e_2 \}\) as bases for \(\displaystyle \mathbb{R}^3\) and \(\displaystyle \mathbb{R}^2\) respectively ...

... BUT ...Poole states that the bases are \(\displaystyle \mathcal{B} = \{ e_1, e_2 , e_3 \}\) and \(\displaystyle \mathcal{C} = \{ e_2, e_1 \}\)... ?So, in the example, it seems that \(\displaystyle T\) is defined in terms of \(\displaystyle \{ e_1, e_2 , e_3 \}\) and \(\displaystyle \{ e_1, e_2 \}\) ... and then we recalculate to find the matrix of \(\displaystyle T\) with respect to \(\displaystyle \mathcal{B}\) and \(\displaystyle \mathcal{C}\) ... ...Can someone explain what is going on here ... shouldn't \(\displaystyle T\) be defined in terms of the declared bases \(\displaystyle \mathcal{B}\) and \(\displaystyle \mathcal{C}\) ... so that \(\displaystyle T\) takes an input in terms of \(\displaystyle \mathcal{B}\), and then give an output in terms of \(\displaystyle \mathcal{C}\) ... ...Hope someone can clarify the above issue ...

Help will be appreciated ..

Peter
 

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  • #2
Note that $\mathcal C$ has the standard unit vectors swapped.
So it is non-standard.
And that's what we see in the matrix of the end result as well. That is, the rows are swapped.
 
  • #3
Klaas van Aarsen said:
Note that $\mathcal C$ has the standard unit vectors swapped.
So it is non-standard.
And that's what we see in the matrix of the end result as well. That is, the rows are swapped.
Thanks Klaas ...

Peter
 

FAQ: Matrices of Linear Transformations .... Poole, Example 6.76 ....

What is a matrix of linear transformations?

A matrix of linear transformations is a rectangular array of numbers that represents a linear transformation between two vector spaces. Each column of the matrix corresponds to a basis vector in the input space, and each row corresponds to a basis vector in the output space.

How is a matrix of linear transformations related to linear algebra?

Matrices of linear transformations are a fundamental concept in linear algebra. They allow us to represent and manipulate linear transformations in a more efficient and organized manner, making it easier to solve complex problems involving vector spaces and linear transformations.

What is the purpose of Example 6.76 in Poole's "Linear Algebra: A Modern Introduction"?

Example 6.76 in Poole's book demonstrates how to find the matrix representation of a linear transformation with respect to different bases. It also shows how to use matrix multiplication to compose linear transformations.

How can matrices of linear transformations be applied in real-world situations?

Matrices of linear transformations have numerous applications in fields such as physics, engineering, computer graphics, and economics. They can be used to model and analyze systems with linear relationships, such as electrical circuits, mechanical systems, and financial markets.

Can matrices of linear transformations be used to solve systems of linear equations?

Yes, matrices of linear transformations can be used to solve systems of linear equations. By representing the coefficients of the equations in a matrix, we can use techniques such as Gaussian elimination to manipulate the matrix and solve the system of equations.

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