MAT 343 Arizona State University Linear Algebra Quiz Questions

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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.
1
2
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
3
(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.