5: Question total, the subject is about (MAT 343) : APPLIED LINEAR ALGEBRA open the file for more information.

QUIZ 3, MAT 343: APPLIED LINEAR ALGEBRA

Instructor: S. K. Suslov

Name:

(1) (10 points) Determine the null space, Ax = 0; of the matrix

0

1

1

2 1

4

1

2

0 A

A=@ 2

2 4

2 8

[Hint: Solve the homogeneous system of linear equations.]

Date: July 23, 2020.

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QUIZ 3, MAT 343

(2) (10 points) Are the vectors v1 = (5; 1; 7); v2 = ( 3; 2; 9); and v3 = (1; 2; 4) linearly

independent? Explain.

(3) (15 points) Let v1 = ( 1; 2; 3)T ; v2 = (3; 4; 2)T ; v3 = (2; 6; 6)T be given vectors in R 3 Is

v3 2 Span(v1 ; v2 )? Prove your answer.

QUIZ 3, MAT 343

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(4) (15 points) Let v1 ; v2 ; v3 be lineary independence vectors in R n and let

u1 = v1 + v2 ;

u2 = v2 + v3 ;

u3 = v3 + v1 :

Are the vectors u1 ; u2 ; u3 lineary independent? Prove your answer.

(5) (Extra credit, 5 points) Let v1 ; v2 ; v3 be lineary independence vectors in R 4 and let A be

a nonsingular 4 4 matrix. Prove that if

u1 = Av1 ;

u2 = Av2 ;

u3 = Av3 ;

then the vectors u1 ; u2 ; u3 are lineary independent.