Linear Algebra question regarding Matrices of Linear Transformations

In summary, the task is to find the matrix representations [T]\alpha and [T]β of a linear transformation T on ℝ3 with respect to the standard basis \alpha = {e1, e2, e3} and β={e3, e2, e1}. T(x,y,z)=(2x-3y+4z, 5x-y+2z, 4x+7y). The solution involves finding the transformation of each basis vector, written as a linear combination of the ordered basis, to determine the columns of the matrix representation. The order of the vectors in the second basis affects the order of the columns in the matrix representation.
  • #1
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Homework Statement



Find the matrix representations [T][itex]\alpha[/itex] and [T]β of the following linear transformation T on ℝ3 with respect to the standard basis:

[itex]\alpha[/itex] = {e1, e2, e3}

and β={e3, e2, e1}

T(x,y,z)=(2x-3y+4z, 5x-y+2z, 4x+7y)

Also, find the matrix representation of [T][itex]^{\alpha}_{\beta}[/itex]

Homework Equations



None

The Attempt at a Solution



T(e1) = (2, 5, 4)

T(e2) = (-3, -1, 7)

T(e3) = (4, 2, 0)

So, I got [T][itex]\alpha[/itex] = (T(e1), T(e2), T(e3))

but for [T]β, I got [T]β=(T(e3), T(e2), T(e1))
However, the answers in the back of the book tell me that although my order for [T]β is correct, the vectors themselves are inverted.
ie: T(e1) = (4, 5, 2)

Why is this? And I'm not sure how to start the second half of the question...
 
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  • #2
The columns of the matrix of a linear transformation in a given ordered basis are the coefficents of the expression of the transformation of each basis vector, in turn, written as a linear combination of that ordered basis.

The reason I emphasised "ordered" is that in the second basis, you are given [itex]\beta[/itex] as same vectors as in [itex]\alpha[/itex], just in a different order.
[itex]T(\beta_1)= T(e_3)= (4, 2, 0)= 0e_3+ 2e_2+ 4e_1[/itex] so the first column is
[tex]\begin{bmatrix}0 \\ 2 \\ 4\end{bmatrix}[/tex]
 
  • #3
Thank you, that cleared up some of my problems!
 

Related to Linear Algebra question regarding Matrices of Linear Transformations

1. What is a matrix of a linear transformation?

A matrix of a linear transformation is a rectangular array of numbers that represents the transformation of a vector space. It consists of rows and columns, with each entry representing the coefficient of a specific vector in the transformation.

2. How do you determine the dimension of a matrix of a linear transformation?

The dimension of a matrix of a linear transformation is determined by the number of columns, which is equal to the number of output vectors in the transformation. For example, a 2x3 matrix represents a transformation from a 3-dimensional vector space to a 2-dimensional vector space.

3. Can a matrix of a linear transformation be invertible?

Yes, a matrix of a linear transformation can be invertible if and only if the transformation is one-to-one and onto. This means that every input vector has a unique output vector, and every output vector has an input vector.

4. How are matrices of linear transformations used in real-world applications?

Matrices of linear transformations are used in a variety of fields, such as computer graphics, physics, and engineering. They can be used to represent transformations of 3D objects, solve systems of linear equations, and model physical processes.

5. Is it possible for two different matrices to represent the same linear transformation?

Yes, it is possible for two different matrices to represent the same linear transformation if they have the same input and output dimensions and the same coefficients for each vector. This is because matrices can be manipulated through row operations without changing the underlying transformation.

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