Math 314H
Solutions to Homework # 1
1. Let = cfw_1 + x, 1 + x2 , x + x2 be a subset of P2 .
(a) Prove that is a basis for P2 .
Let = cfw_1, x, x2 be the standard basis for P2 and consider the linear transformation T : P2 R3 dened by T (f ) = [f ] , wher
Math 314H
EXAM I
Do all of the problems below. Point values for each of the problems are adjacent to the
problem number. Calculators may be used to check your answer but not to arrive at your
answer. That is, you must show all of the work necessary to der
Math 314H Quiz 7 Solutions
3
1. Let V = R3 with the dot product as its inner product. Let u = 4. Find
5
u and the unit vector in the direction of u.
Solution: u = 32 (4)2 + (5)2 = 50. The unit vector in the direction
+
of u is
1 u
50
=
3
50
4
50 .
5
5
Math 314H Quiz 5
For each of the following decide whether the subset W is a subspace of the given
vector space V . In each case prove your answer is correct.
1. Let V = R (i.e., the vector space of innite sequences of real numbers) and
W = cfw_(a0 , a1 ,
Math 314H Quiz 4 Solutions
The row reduced echelon form of
11 2 1 0
1 4 1 4 6
A=
2 5 1 5 6
1 1 2 6 5
is the matrix
1
0
0
0
03
1 1
00
00
02
0 1
1 1
00
1. Find a basis for Nul A and nd dim Nul A (i.e., the nullity of A).
Nul A is the set of all solutions to
Math 314H Solutions to Quiz 3
Let T : R3 R4 be the linear transformation dened by
x1 3x3
x1
x2
x2 =
T
x1 + x2 3x3
x3
x1 2x2 3x3
1. Find the standard matrix for T .
Let cfw_e1 , e2 , e3 be the standard basis for R3 . Then
1
0
0
1
T (e1 ) =
T (e2 )
Math 314H Solutions to Quiz 2
Consider the following 4 4 matrix:
1 1 3
0
1
5
A=
1 2
8
3 1 1
0
4
5
3
The row reduced echelon form of A is
1
0
R=
0
0
0
1
0
0
2
5
0
0
0
0
1
0
1. What is the general solution to the system Ax = 0?
The general solution is:
x1
x
Math 314H Solutions to Quiz 1
Consider the following system of equations in 3 variables:
x + 2y 7z = 0
2x + 3y +9z = 4
2y +10z = 8
1. Write down the augmented matrix for the system.
The augmented matrix for this system
1
2
0
is:
2 7 0
3 9
4
2 10 8
2. Put
Math 314H
Solutions to Homework # 3
1. Complete the exercises from the second maple assignment, which can be downloaded from
my linear algebra course web page. Attach printouts of your work on this problem to your
solutions to the rest of the problems bel
Math 314H
Solutions to Homework # 2
1. Let T : U V be a linear transformation. Suppose cfw_u1 , . . . , uk is a basis for ker T
and cfw_v1 , . . . , v a basis for im T . Now, as v1 , . . . , v are in im T , there exists vectors
w1 , . . . , w in U such