Math 150-01 Linear Algebra Homework II Solutions
1. Decide if the following subsets of R2 are vector spaces, and justify your answer.
(a) S = cfw_(x, y ) | x + y = 1, the set of all points in the xy -plane whose coordinates add
to 1
S is not a subspace. I
Math 150-01 Linear Algebra
Exam Review Sheet
Exam II: October 28, 2013
You will be expected to know:
what a vector space is, and the ten axioms needed to show a set is a space
what a subspace is, and the two axioms needed to show a subset is a subspace
Math 150-01 Linear Algebra Exam 1 Review Solutions
1. Compute the length of the vector v = [1, 4, 2, 3]
|v| =
12 + 42 + (2)2 + 32 =
30
2. Compute the dot product u v when u = [1, 2, 0, 2] and (a) v = [1, 2, 8, 1], (b) v =
[8, 3, 6, 1]
(a) u v = (1)(1) + (
Math 150-01 Linear Algebra
Exam Review Sheet
Exam I: September 30, 2013
You will be expected to know:
what a vector is, and how to represent one geometrically
how to add and subtract vectors, and when addition is dened
what a linear combination of vect
Math 150-01
Exam III
November 22, 2013
Name:
Show all work for full credit. Partial credit may be given if merited.
1. Find the determinants of the following matrices. If you do not calculate the determinant
directly (via cofactor expansion, for example),
Math 150-01 Linear Algebra Homework I Solutions
1. When working with numbers, we know that multiplication distributes across addition:
a(b + c) = (b + c)a = ab + ac
The dot product is also a distributive operation, across vector addition. If v, u, and w
a
Math 150-01
Exam I
September 30, 2013
Name:
Show all work for full credit. Partial credit may be given if merited.
1. Compute the following, or explain why the operation is not dened.
62
1
(4)
(a) 1 3
2
04
(3)
1
(b) 4 6 0 0
9
(4)
(c)
(4)
1 4
3 1
(d) 2 6
Math 150-01
Exam II
October 28, 2013
Name:
Show all work for full credit. Partial credit may be given if merited.
(5) 1. Show that the set of vectors
V=
x
x
| x is a real number
is a subspace of R2 .
2. For each of the sets below, provide one reason why t
Math 150-01 Linear Algebra Homework II Solutions
1. Multiplying a matrix A on the left by a permutation matrix Pij interchanges the rows i
and j of the matrix A. Construct a 3 3 matrix A, and nd the product APij for various
permutation matrices Pij . What
Math 150-01 Linear Algebra Homework II Solutions
1. Solve the following systems of equations by row reducing the augmented matrix representing the system.
2x + y 2z = 10
(a) 3x + 2y + 2z = 1
5x + 4 y + 3 z = 4
We need to row reduce the augmented matrix [A
Math 150-01 Linear Algebra Homework V Solutions
1. Find the determinant of each of the following matrices. You can use any method you
prefer, but you must show all work for full credit.
(a) A =
2 5
03
2 5
= (2)(3) (0)(5) = 6 0 = 6
03
1 8 0
(b) A = 0 2 4
3
Math 150-01 Linear Algebra Homework VII Solutions
x
y
1. Dene a transformation T : R2 R3 by T
x+y
= x 2y .
3x
(a) Show that this is a linear transformation by verifying the axioms. For full credit,
your work should be as general as possible (for instance,
Math 150-01 Linear Algebra Homework VI Solutions
1. Let
1
2
1
A = 6 1 0
1 2 1
(a) Find the eigenvalues of the matrix A.
The characteristic polynomial is
1
2
1
1
0 = 3 2 + 12 = ( 3)( + 4)
det 6
1
2
1
so the eigenvalues are 0, 3, and -4. (Note, you reall
Math 150-01 Linear Algebra Exam 2 Review Solutions
1. Consider the set of vectors
a
S = b : a = b
c
That is, S is the set of vectors with three components, where the rst two components
are equal. Decide whether or not S is a subspace of R3 .
We already