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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 :

4

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.

8

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 :