MATH 0280 - Review 3
1. Review 1.
2. Review 2.
3. Show that A and B are not similar matrices:
3 1
A=
,
5 7
4. Determine whether A is diagonalizable,
D such that P 1 AP = D.
1
A = 0
1
5. Section 5.2:
3, 7, 9, 14, 17, 21.
6. Section 5.3:
4, 6, 8, 10, 16 , 1
Written Assignment 6 - ANSWERS
1) (All solutions are listed as horizontal vectors)
a) Unique solution [1, 2].
b) Unique solution [5/6, 2/3, 7/6].
c) No solutions.
d) Unique solution [2, 1/5, 0, 4/5].
2) a) A1 =
1 2
1
2
1
2
b) Not invertible.
1 0 2
1
6
c
QUIZ # 4 (TAKE-HOME)-ANSWERS
Problem 1. x = 4, y = 5, z = 11.
Problem 2.
a) A1
1/2 1
0
0
0 ,
= 1/2
1/2 1/3 1/3
b) A is diagonalizable, since it has 3 distinct eigenvalues.
3
0 0
0 2
1
D = 0 1 0 , P = 0 1 1 .
0
0 2
1
1
1
(Note: D and P are not unique)
c) A
NAME:
Written Assignment # 2
Due Date: Monday, February 9th, at the beginning of the lecture
Instructions: Please write your work on a separate paper, and attach
this page with your name to the top. Make sure to show all your work.
Please keep the copy of
Written Assignment # 3 - ANSWERS
1) a)
2 3 6 3
1 1 2
2
1 1 2
1
5 7 15 8
b)
26
13
9
68
2) NOTE: the matrix representations provided are not unique. To check
if the problem is solved correcly, multiply the matrices you obtained and
compare the resul
Written Assignment # 4 - ANSWERS
1 a) dim(row(A) = 3, a possible choice for the basis: [2, 0, 2, 0, 2], [0, 2, 2, 0, 3], [0, 0, 0, 1, 0].
b) dim(col(A) = 3, a possible choice for the basis (column vectors are
written horizontally here): [2, 0, 0, 0], [2,
NAME:
Written Assignment # 1
Due Date: Wednesday, January 21st, at the beginning of the lecture
INSTRUCTIONS: Please write your work on a separate paper, and
attach this page with your name to the top. Make sure to show all your
work. Please keep the copy
MATH 0280 Final Examination, Sample 4-ANSWERS
Problem 1. A has eigenvalues = 0 with alg. and geom. multiplicities
2, and = 1 with alg. and geom. multiplicities 1.
A2012
4
2
2
1
1
= 2
8 4 4
2
23
Problem 2. a) A =
1
8
b) A1 =
3
8
3
2
1
2
3
2
1
2
.
Proble
MATH 0280 - Review 1
1. For the given two vectors:
u=
3
,
1
v=
1
,
2
Find the angle between u and v, and find the projection of v onto u.
2. Find all values of scalar k such that the following two vectors are orthogonal:
2
k+1
u=
, v=
.
3
k1
3. Use
MATH 0280 - Review 2
1. Find the bases of row(A), col(A), and null(A), as well as rank(A), rank(A> ), and nullity(A) for the
following matrices:
1
A = 0
0
1
1
1
0
1
1
1
1 ,
1
2. Find a basis for the span of the following vectors:
1
1
1 , 0 ,
0
1
2
A =
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MATH 0280 Final Examination, Sample 4
Problem 1. Show that A is diagonalizable. Then nd A2012 .
4
2
2
1
1
A= 2
8 4 4
Problem 2. a) Find the standard matrix of the linear transformation
T : R2 R2 , if T rotates a vector clockwise about the origin by 23 , a
MATH 0280 Final Examination, Sample 3-ANSWERS
Problem 1. a) = 0(alg. mult. 1); = 1 (alg. mult.2)
b) For = 0 : Basis of E : [1/2, 1/2, 1] (vertical vectors are written
horizontally here and below to save space)
For = 1 : Basis of E : cfw_[2, 1, 0], [1, 0,
MATH 0280 Final Examination, Sample 3
Problem 1.(30 pts) Matrix A is given as
0 2 1
A = 1 3 1
2 4 1
a)(10 pts) Find all eigenvalues of A.
b)(10 pts) Find a basis for each eigenspace of A.
c)(10 pts) Determine whether A is diagonalizable. If it is, nd an