MAT 343 Arizona State University Applied Linear Algebra Exam

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FINAL EXAM, MAT 343: APPLIED LINEAR ALGEBRA
Instructor: S. K. Suslov
Name:
Show your work!
(1) Let
(a) (25 points) Find A 1 :
0
7
A=@ 0
3
2
3
4
1
1
1 A:
2
(b) (15 points) Solve the system Ax = (1; 1; 1)T :
Date: August 11, 2020.
1
2
FINAL EXAM, MAT 343
(2) (15 points) Evaluate the following determinant:
0
1
1 2 3
1 1 A
det @ 1
3 2 1
(3) (15 points) Let A and B be n n similar matrices, namely, B = S 1 AS: Show that the
matrices A and B have the same characteristic polynomial, det (A
I) = det (B
I) ;
and, consequently, the same eigenvalues.
FINAL EXAM, MAT 343
3
(4) (25 points) Convert the basis v1 = (1; 1; 0); v2 = (0; 1; 1); v3 = (1; 1; 1) for R 3 into an
orthonormal basis, using the Gram{Schmidt process and the standard inner product in R 3 :
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FINAL EXAM, MAT 343
(5) Find the eigenvalues and associated eigenvectors of a given matrix A:
2 1
(a) (15 points) A =
6
3
0
2
@
3
(b) (25 points) A =
1
1
0 0
7 2 A
2 2
FINAL EXAM, MAT 343
(6) (25 points) Solve the following system of linear equations:
x+y z w =2
x y z+w =6
:
x + y 3z 2w = 4
x + y + 2z + w = 1
5
6
FINAL EXAM, MAT 343
(7) (25 points) Find both a basis for the
matrix
0
2
@ 2
0
row space and a basis for the column space of the
4 2
1 1
3 3
1
8
0 A
8
What is the rank of this matrix? Find the nullspace basis of this matrix.
FINAL EXAM, MAT 343
7
(8) (15 points) The linear transformation, L : R 3 ! R 2 is given by
L (x1 ; x2 ; x3 ) = (x1 + x2
x3 ; x1
x2 + x3 ) :
Find the matrix representation of L with respect to the canonical bases.
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FINAL EXAM, MAT 343
(9) (Extra Credit, 10 points) Show that for any two vectors u and v in an inner product space
V;
ku + vk2 + ku
vk2 = 2 kuk2 + kvk2 :
Give a geometric interpretation of this result for the vector space R 2 :