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QUIZ 4, MAT 343: APPLIED LINEAR ALGEBRA

Instructor: S. K. Suslov

Name:

(1) (15 points) Find a basis for the row

null space of the following matrix:

0

1

@ 2

3

space, a basis for a column space, and a basis for the

3

1

4

2

3

5

What is the rank of this matrix?

Date: August 4, 2020.

1

1

1

2 A

3

2

QUIZ 4, MAT 343

(2) (15 point) Convert the basis v1 = ( 1; 1; 0)T ; v2 = (2; 1; 1)T ; v3 = (0; 1; 1)T for R 3 into

an orthonormal basis, using the Gram{Schmidt process and the standard inner product in

R 3 : Expand the vector v4 = (1; 2; 3)T with respect to this new basis.

QUIZ 4, MAT 343

(3) (10 point) Find

0

1

B 2

det B

@ 1

0

3

8

3

3

2

6

3

2

3

1

0

2 C

C:

1 A

2

4

QUIZ 4, MAT 343

(4) (10 points) Find the angle between the vectors u = ( 1; 2; 3; 1) and v = (1; 3; 2; 2) and

verify the Cauchy{Schwarz inequality, jhu; vij

jjujj jjvjj ; for them using the standard

inner product in R 4 : hu; vi := u1 v1 + u2 v2 + u3 v3 + u4 v4 :

(5) (Extra credit, 5 points) Show that

Z b

(f; g) =

f (x) g (x)

a

(x) dx;

(x) > 0

is an inner product in C [a; b] (the space of continuous functions on [a; b]). Prove the corresponding triangle inequality.