QuestionFind the standard matrix for the linear transformation that rotates points in by radians. (Enter exact values for each matrix element.)
Studdy Solution
STEP 1
Assumptions1. is a linear transformation.
. rotates points about the origin through radians.
3. We need to find the standard matrix of.
STEP 2
The standard matrix of a linear transformation is the matrix for which(v) equals multiplication by A for each vector v in the domain. In other words, if is a linear transformation from to , then there is a unique m x n matrix A such that(v) = Av for all v in .
STEP 3
To find the standard matrix of, we need to apply the transformation to the standard basis vectors in , which are and .
STEP 4
First, apply the transformation to the first standard basis vector . Since rotates points about the origin through radians, the result is .
STEP 5
Next, apply the transformation to the second standard basis vector . Again, since rotates points about the origin through radians, the result is .
STEP 6
The standard matrix A of the linear transformation is then given by the column vectors obtained from the previous steps.
So, the standard matrix of the linear transformation that rotates points about the origin through radians is .
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