24-677 (LCS) Homework 3 solution

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1: 20 points
Consider two bases for space R
3
.
{a1, a2, a3} =





2
1
4

 ,


3
−2
−2

 ,


4
2
1





, {b1, b2, b3} =





−2
3
1

 ,


−4
−3
−2

 ,


5
−2
0





If a linear operator is given in the {a1, a2, a3} basis by
A =


8 −2 −1
4 −2 −3
2 −3 −3


(a) Find the representation for this operator in the basis {b1, b2, b3}.
(b) If we are given a vector x =

2 −1 −4
T
in the basis {a1, a2, a3}, determine the
representation for x in the basis {b1, b2, b3}.
2: 10 points
For the pair of matrices
A =




2 3 1 4 −9
1 1 1 1 −3
1 1 1 2 −5
2 2 2 3 −8




, y =




17
6
8
14




Determine all the possible solutions to the system Ax = y.
3: 10 points
Find the best solution, in the least-squared error sense, to the equations
−2 = x1 − 2×2
5 = x1 − 2×2
1 = −2×1 + x2
−3 = x1 − 3×2.
4: 15 points
Find the eigenvalues and corresponding eigenvectors of the following matrices.
1