Name:
Date: 6-27-08
Exam 2
Remember to show all work. No calculators, notes, or books are allowed on this Quiz. When answering
True/False questions write the entire word. No proof is needed. A statement is True if it is true without
exception. Circle all
Math461 Midterm 2
Prof. Konstantina Trivisa
March 31, 2016
Instructions: Read each problem carefully. Write clearly and show all your work in your
notebook. You may NOT use calculators.
Problem 1. (30 points)
Assume that the matrix A, is row equivalent to
MATH461 - Midterm 2 Solutions
Problem 1
(a) To find a basis for the null space, solve Ax = 0. From the row reduced matrix, this simplifies to
x1 2x2 x4 = 0,
x3 + 2x4 = 0.
Our free variables are x2 and x4 . In vector form, the solution is
x1
2x2 + x4
1
Linear Algebra and Its Applications, Fifth Edition, Lay, Lay, & McDonald
Chapter 1 Linear Equations in Linear Algebra
1.1 Systems of Linear Equations:
A linear equation in the variables
a1 x1 +a2 x 2 +a n x n=b
x1 , x2 , xn
is an equation that can be wr
Homework 5
Math 461
Due Tuesday, October 21st.
Problem 1 Let A be the matrix
1 4
1
3
0 0
7
7
A=
3 12 18 12 .
2 8 5 1
Find a basis for the column space and null space of A.
Solution: A in row reduced form is given by
1
0
U =
0
0
4
0
0
0
1
7
0
0
3
7
.
Homework 3
Math 461
Due Tuesday, September 30th.
Problem 1 Let A be the matrix
4
2
3
A = 16 6 17 .
12 16 10
Complete a LU -decomposition of A. Recall, you nd a sequence of row operations R1 , . . . , RN
1
1
such that RN R1 A is in row echelon form. Then c
Homework 1
Math 461
Due Tuesday, September 23rd.
Problem 1 Consider the matrices,
A=
1 2
B = 1 6 .
0 1
1 0 2
1 1 3
Calculate AB. For any vector y = (y1 , y2 )t , nd a vector x R3 such that Ax = y.
Problem 2 Consider the matrices,
A=
14 6
0 1
B=
12 5 .
Homework 1
Math 461
Due Tuesday, September 16th.
Problem 1 Consider the matrix
cos sin
sin cos
U=
.
Let v, w be any 2-dimensional vectors. Show that (U v) (U w) = v w. Also, show that
|U v| = |v|
Any square matrix A where (Av) (Aw) = v w holds for all v
Homework 4
Math 461
Due Tuesday, October 14th.
Problem 1 Let A be the matrix
1 4 0
A = 2 8 1 .
5 1 13
Complete a LU -decomposition with partial pivoting of A. That is, nd a lower triangular
L, upper triangular U , and permutation matrix P such that P A =
Homework 5
Math 461
Due Tuesday, October 21st.
Problem 1 Let A be the matrix
1 4
1
3
0 0
7
7
A=
3 12 18 12 .
2 8 5 1
Find a basis for the column space and null space of A.
Problem 2 Let A be an m n matrix. When m > n, prove there exist some b Rm such
t
Homework 6
Math 461
Due Tuesday, October 28th.
Problem 1 Are the following vectors linearly independent?
0
2
3
3
v1 = 3 , v2 = 3 ,
v3 =
3
1
2
2
2
0
3
2
2
.
Problem 2 Are the following vectors linearly independent? (Hint: There is no need to
perfo
Homework 7
Math 461
Due Tuesday, November 4th.
Problem 1 Project b onto the column space of A where,
1 1
4
A= 1 1
b = 2 .
0 1
8
Problem 2 Let
2 3
A = 0 3
0 0
1
b = 2 .
5
Solve for x where |A b| = minxR2 |Ax b|.
x
Problem 3 Let V be the subspace spanne
Math 461/Fall 2013/Jerey Adams
Review for Exam 2
1. Chapter 2, Sections 2.8 and 2.9
2.8 Subspaces of Rn : denition of subspace, column space, null space,
basis of a subspace
2.9 Dimension and Rank: coordinates, dimension of a subspace, the invertible matr
Math 461/Fall 2013/Jerey Adams
Test I/Monday October 7, 2013
Calculators are allowed but not needed
and you must show all work for full credit
#1 [27 points]
112
(a) Find the reduced row echelon form of A = 1 2 3
235
1
1
1
(b) Find the general solution to
Math 461/Fall 2013/Jerey Adams
Test III/Wednesday December 11, 2013
Each problem worth 20 points
Problem 1. Suppose A =
1
1
1
.
3
(a) Find the eigenvalues of A.
(1 )(3 ) + 1 = 2 2 + 3 + 1 = 2 2 + 4 = ( 2)2 . So there is
only one eigenvalue 2.
(b) For each
Math 461/Fall 2013/Jerey Adams
Test II/Wednesday November 6, 2013
SOLUTIONS - Each problem worth 20 points
Problem 1.
(a) Suppose A is a nonzero 1 5 matrix. What is its rank?
Solution: The rank is the dimension of the column space, which is a subspace
of
Math 461/Fall 2013/Jerey Adams
Solutions to Homework 7, Due 10/22
Section 4.1, #2, 6, 10, 16, 24; Section 4.2, #4, 8, 16, 26; Section 4.3, #8, 10,
16, 20, 22
Section 4.1, #2
(a) Yes. Look at the picture. Or note that if v = (x, y ) and xy 0 then
cv = (cx,
Math 461/Fall 2013/Jerey Adams
Solutions to Homework 2
Section 1.4, #2, 4, 5, 12, 16, 19, 22; Section 1.5, #2, 5, 6, 12, 17, 18, 24;
Section 1.7, #5, 6, 12, 16, 22
#2. Not dened, the number of columns of the matrix (1) is not equal to the
number of rows o
Math 461/Fall 2013/Jerey Adams
Solutions to Homework 5, Due 10/4
Section 2.5 #2, 4, 6, 8, 10
#2. First solve Ly = b, i.e.
1 00
2
x
2 1 0 y = 4
0 11
6
z
2
So x = 2, 2x + y = 4 y = 2x 4 = 0 and y + z = 6 so z = 6. So y = 0.
6
Now solve U x = y :
2 6 4
x
Math 461/Fall 2013/Jerey Adams
Solutions to Homework 5, Due 10/4
Section 2.8, #2, 6, 8, 12, 16, 22; Section 2.9, #4, 6, 8, 15, 16, 18; Section 2.6,
#4, 6, 7, 8
Section 2.8, #2. Take v in the upper left hand corner, and w in the lower
right. Then v + w is
LU Factorization
One-to-one and onto
Rank, dimension, row space, col space, null space
Bases, change of basis
Determinants, Changing area by determinants
Difference equations
Diagonalization
Discrete dynamical systems
Differential equations
Projections
Be