Math 3130 - Intro to Linear Algebra - MathOnline -...

advertisement
Math 3130 - Intro to Linear Algebra - MathOnline - Spring 2016
Instructor: Dr. Radu C. Cascaval
Department of Mathematics - University of Colorado at Colorado Springs
Email: [email protected], Fax: 1-888-951-4321 (toll free)
To be read and completed by the Proctor: By signing below, I certify that the exam was administered in the time allowed (75 minutes) and observing the following rules: no
books/notes; scientific calculator (WITHOUT graphic or matrix capabilities) is allowed.
Proctor Name:
Proctor Signature (required)
(Please fax or email the completed exam no later than Wed, April 20, 5pm MT.)
Student Name:
Exam 2 - Math 3130 - Intro to Linear Algebra
The exam has 8 problems on 5 pages. Make sure you turn in ALL pages. To receive credit, show all work.
No calculators with linear algebra capabilities allowed!
Exercise 1.[3 pts each]
Determine whether each of the following statements are TRUE or FALSE.
Circle your answer and justify it for credit. No justification, no credit.
• The set of nonzero row vectors of a matrix A is a basis for the row space of A
TRUE
FALSE
• The reduced row echelon form of AT is the transpose of the reduced row echelon form of A.
TRUE
FALSE
• Every linearly independent subset {v1 , . . . , vn } of a n-dimensional vector space is a basis.
TRUE
FALSE
• The span of any finite set of vectors in a vector space V is a subspace of V .
TRUE
FALSE
Exercise 2.[5 pts each]
Decide whether Cramer’s rule can be used in solving each of the linear system below or not. If yes, solve it using
Cramer’s rule. If not, explain why.
(
3x1 2x2 = 1
(a)
x1 2x2 = 2
8
>
< x + 2y + 3z
(b) x y + z
>
:
x + 5y + 5z
=1
=2
=0
Exercise 3.[5 pts each]
(a) Find the complete solution of the linear system
2
1 2
A = 42 6
0 2
Ax = b, where
3
2 3
1
2
35 ,
and b = 445
1
0
(b) What is the rank of the matrix A? What is the column space of A?
Exercise 4.[5 pts each]
(a) Find a basis for the subspace of R3 spanned by the following 4 vectors
2 3
2 3
2 3
2 3
1
0
2
1
v1 = 4 35 , v2 = 415 , v3 = 4 7 5 , v4 = 405
0
2
2
1
(b) Express all the remaining vectors as linear combinations of the basis found in (a).
Exercise 5.[9 pts]
Find a basis for the subspace of R4 consisting of all vectors that are orthogonal to
v1 = (1, 1, 2, 1)T and v2 = (0, 0, 1, 3)T
.
Exercise 6.[6 pts each]
Given the matrix
(1) Determine the rank of A.
2
1
A = 41
0
(2) Find a basis for the column space of A.
(3) Find a basis for the row space of A.
(4) Find a basis for the nullspace space of A.
2
2
0
1
0
1
4
5
1
3
1
05
1
Exercise 7.[5 pts each]
Find a basis and dimension of the following subspaces of R4
(a) All vectors of the form (a, b, c, d) with a + b + c + d = 0
(b) All vectors of the form (a, b, c, d) with a = b = c = d.
Exercise 8.[5 pts each]
Determine which of the following are subspaces of Mn⇥n , the space of all n ⇥ n matrices and which are not. Find
the dimension of these subspaces.
(a) The set of all diagonal n ⇥ n matrices.
(b) The set of all n ⇥ n matrices A such that det(A) = 0.
(c) The set of all symmetric matrices A (AT = A)
Download