Math 441
Exam 1
x1
x2
1. (8 pts) a) Given a vector x
x2
x3
find A so that Ax
0
1
1
0 0
0
x2
with A the indicated n
1
n matrix.
xn
xn
1
x1
x2
1
as a linear combination
xn
x3
xn
x1
x3
b) Express the v
Math441, 2017S Homework #1. Name:
Due date is 1/25/17. Show your work.
1. Consider the system
x + dy + z = 0,
2x + 4y + z = 1,
x + y + z = 0.
(a) Construct the augmented matrix respecting the order of
Math441, 2017S Homework #2. Name:
Due date is 2/1/17. Show your work.
1. The elimination algorithm was applied to the matrix A. Suppose it yields the following
equality to A.
2 1 1
1 0 0
1 0 0
1 0 0
Fred and Carries Research Report
Washing Machine and Formula Information
Fred and Carrie are thinking of purchasing washing machines that have a 20-pound capacity, meaning
that each machine can hold 2
Solutions to Exercises
27
Problem Set 3.1, page 127
1 x C y y C x and x C .y C z/ .x C y / C z and .c1 C c2 /x c1 x C c2 x .
2 When c.x1 ; x2 / D .cx1 ; 0/, the only broken rule is 1 times x equals x
Solutions to Exercises
2
Problem Set 1.1, page 8
1 The combinations give (a) a line in R3
2
3
4
5
6
7
8
9
10
11
(b) a plane in R3 (c) all of R3 .
v C w D .2; 3/ and v w D .6; 1/ will be the diagonals
Solutions to Exercises
z2 z1 D b1
10 z3 z2 D b2
0 z3 D b3
7
z1 D
z2 D
z3 D
b1
b2
b2
b3
b3
b3
D
"
1
0
0
1
1
0
1
1
1
#"
b1
b2
b3
#
D
1
b
11 The forward differences of the squares are .t C 1/2
t 2 D t 2
Solutions to Exercises
42
23 As in Problem 22: Row space basis .3; 0; 3/; .1; 1; 2/; column space basis .1; 4; 2/,
24
25
26
27
28
29
30
31
32
.2; 5; 7/; the rank of (3 by 2) times (2 by 3) cannot be l
Math 441
Exam 3
1. (8 pts) Subspaces of R n are generally described either as the column space of some
given matrix, or the null space of some given matrix.
a) If we define a subspace V as the nullspa
The four subspaces as orthogonal complements
Calculating basis for an orthogonal complement - How? Calculating orthonormal basis for
an orthogonal complement - How?
Ax b has a solution, b is in the co
Math 441 Exam 4
5
6
3
1. (12 pts) Given A
4
a) Find the eigenvalues and eigenvectors of A
5
6
3
det
4
2
2 0 , 1, 2
1:
A
6
6
3
3
3
6
3
I
,v
6
1
,v
2
1
2:
A
I
1
b) Write A in diagonalized form, A
Math 441
Exam 4
Eigenvalues/eigenvectors:
1) I give you a matrix and you find the eigenvalues and eigenvectors
2) I tell you an eigenvalue and ask you to find the corresponding eigenvectors (basis of
Math 441 Exam 2
1. (2.5 pts each, total 15) Answer the following simple questions, and provide a brief
explanation.
a) If the vectors in the set S v 1 , . . , v k are linearly independent, what is a b
MATH 441.001
Instr. K. Ciesielski
Spring 2011
NAME (print):
FINAL TEST Review
Final Test will start with: Solve the following exercises. Show your work. (No credit will
be given for an answer with no
1. a) Notice that the second vector is three times the first, so all linear combinations of the
two vectors can be expressed as a scalar multiple of the first vector. So the result is the line
t 1, 2,
Sec. 1.2
3,4,5,6,8,9,11,12,16,17,21,23
v 1 3, 4 and w 1 8, 6 give unit vectors in the directions of v, w
5
10
v
w
v w v w 1 3, 4 1 8, 6 48 24 , and
Then cos
5
10
50
25
vw
v
w
1 24
cos
0. 283 79 (in
Math 441
Review topics for first exam
Vectors, linear combinations of vectors
Ax as a linear combination of the columns of A, as the dot product of x with the rows of A
Solve Ax b
Express b as a linea
General ideas:
Linear independence, span, basis dimension
Linear independence: c 1 v 1 . . . c k v k 0 only when c 1 0 , . . . , c k 0
Equivalent statements (if v s are column vectors):
Ax 0 only if x
Math441-17S Exam #1. 100pts Total. Name:
Show your work. The solutions without work will not get any credit.
1. If the elimination is applied to Ax = b, we obtain the augmented matrix on the right.
1