MATH 343 ASU Applied Linear Algebra Quiz

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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.