Matrix Representation of a linear transformation Similarity Application: Computer Graphics __see att

Matrix Representation of a linear transformation

Similarity

Application: Computer Graphics

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Matrix Representation of a Linear Transformation
T: Rn ? Rn is a linear operator
Find the matrix representation for T relative to the standard basis for Rn
Find T(v), using a direct computation and using the matrix representation.
T: R3R3
T =
v =
T: V ? V is a linear operator with B and B’ ordered bases for V
Find the matrix representation for T relative to the ordered bases B and B’
Find T(v), using a direct computation and using the matrix representation.
T: R3R3
T =
B =
B’ =
v =
Find
Let T: R2 ? R3 be the linear transformation defined by
T =
Let B and B’ be ordered bases for R2 and B” the ordered basis for R3 defined by
B =
B’ =
B” =

Find the matrix representation for the given linear operator relative to the standard basis.
Compute the image of v = directly and using the matrix found in part (a).
Let S, T: R2R2 be defined by
T = andS =
Similarity
A linear operator T and bases B1 and B2 are given.
Find [T]B1 and [T]B2
Verify that the action on v of the linear operator T is the same when using the matrix representation of T relative to the bases B1 and B2
T =
B1 =
B2 =
v =
[T]B1 and [T]B2 are, respectively, the matrix representations of a linear operator relative to the bases B1 and B2. Find the transition matrix P = and show directly that the matrices are similar.
[T]B1 = [T]B2 =
B1 =
B2 =
Find the matrix representation of the linear operator T relative to B1. Then find [T]B2
T =
B1 =
B2 =
Application: Computer Graphics
Let T: R2 ? R2 be the transformation that performs a reflection through the y-axis, followed by a horizontal shear by a factor of 3.
Find the matrix of T relative to the standard basis.
Let T: R2 ? R2 be the transformation that performs a reflection through the line y = x, followed by a rotation of 90 degrees.
Find the matrix of T relative to the standard basis.

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