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Download EXAM 2 Prof. Alexandru Suciu MTH 1230 LINEAR ALGEBRA
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Transcript
Prof. Alexandru Suciu
MTH 1230
LINEAR ALGEBRA
EXAM 2
1. 12 points
1 2
Let A = −1 0
2 3
(a) Find a basis for
3
1
0
the
4 5
2 3.
5 8
image of A.
(b) Find a basis for the kernel of A.
(c) Find the rank and the nullity of A.
Spring 2001
MTH 1230
Exam 2
2. 16 points Consider the folowing four vectors in R4 .
1
0
0
0
2
1
v1 =
v2 =
v3 =
1 ,
0 ,
3 ,
0
3
3
Spring 2001
2
1
v4 =
7 .
4
Also let A be the 4 × 4 matrix with columns v1 , v2 , v3 , v4 .
(a) Are the vectors v1 , v2 , v3 , v4 independent or dependent? If they are independent, say why.
If they are dependent, exhibit a linear dependence relation among them.
(b) Do the vectors v1 , v2 , v3 , v4 form a basis for R4 ? Explain your answer.
(c) Does the equation A · x = 0 only have the solution x = 0, or does it have other solutions?
Explain your answer.
(d) Does the equation A · x = b have a solution for every choice of b in R4 ? Explain your
answer.
MTH 1230
Exam 2
Spring 2001
3. 10 points Let V be the subspace of R3 defined by the equation x1 + 2x2 − 5x3 = 0.
(a) Find a basis for V .
(b) Find a linear transformation T : R2 → R3 such that ker T = {0} and im T = V . Describe
T by its matrix A.
MTH 1230
Exam 2
4. 12 points In each of the following, a subset V of R3 is given. Circle one answer:
x+y+z
(a) V = x + z
y
x, y, z arbitrary constants
Is closed under addition:
YES NO
Is closed under scalar multiplication: YES NO
YES NO
Is a vector subspace of R3 :
x+y+z
(b) V = x + z
y+1
x, y, z arbitrary constants
Is closed under addition:
YES NO
Is closed under scalar multiplication: YES NO
YES NO
Is a vector subspace of R3 :
x
(c) V = y
z
x, y, z positive integers
Is closed under addition:
YES NO
Is closed under scalar multiplication: YES NO
YES NO
Is a vector subspace of R3 :
x
(d) V = y
z
xy ≤ 0
Is closed under addition:
YES NO
Is closed under scalar multiplication: YES NO
YES NO
Is a vector subspace of R3 :
Spring 2001
