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Unformatted text preview: present any linear
transformation as a matrix.
a. Find the matrix representation of linear transformation that does the rotation of the
vector about the origin by 90o clockwise. That is, find such that
is the vector
o
rotated 90 about the origin clockwise.
b. Find the matrix representation of the transformation which does rotation about the
origin for an arbitrary angle, . Solution
a.
b. ([ ])
([ ]) [
[ ]
[ ()
() ()
() 4. Linear Transformations
Consider Example 3 on page 23. Find the P matrix and verify that
Need to solve ̅
[ Solving this similar to problem 1 gives 7 December 2012 [ [ 6 ̅ . . Dept. of ECE, Drexel University ECES511: Systems I Fall 20122013 5. Linear Transformations – Example 4 pg. 24
You may use MATLAB or some other numerical tool to assist you in the computations.
Consider the following matrix representation of a linear operator:
[ a. Find the rank and nullity of the matrix A.
b. Choose an appropriate vector b to be used to compute the companion form of the matrix A.
(Hint: Choose something simple as Example 4 did.)
c. Verify that
qualify to form a basis. What condition must be satisfied?
d. Write the representation of w.r.t. the basis
.
e. Find the change of basis matrix Q and verify that ̅
. Solution
a. Rank = 4, Nullity = 0
[ b.
c. [ [ , which has rank 4. Therefore all columns are linearly independent and this qualifies as a basis. d. [ e. Same as part c. 7 December 2012 7 Dept. of ECE, Drexel University...
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