Question
For the linear transformation
given by
find the matrix
for
with respect to the bases
Ask by Mitchell Sherman. in Russia
Dec 16,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
The matrix
for the linear transformation
is:
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To find the matrix
for the linear transformation
with respect to the bases
and
, we follow these steps:
-
Write the Transformation: The linear transformation is given by
-
Compute the Images of the Basis Vectors: We will apply
to each vector in the basis .- For
: - For
:
- For
-
Express Images in Terms of the Basis
: Next, we express the images in terms of the basis .-
For
:We set up the equations:-
-
-
(since the third component is 0)
From the third equation, . Substituting into the second equation gives:Substituting into the first equation:Therefore, in terms of the basis : -
-
For
:Setting up the equations:From the third equation, . Using this in the second equation gives:Using this in the first equation:Therefore, in terms of the basis : -
-
-
Construct the Matrix: The resulting vectors from the transformation can be arranged as columns in the matrix:[
A_{