Math 113 Homework 9
Not due (these are suggested problems only)
This assignment is a collection of practice problems for the most recent material, as well as
some earlier problems in order to practice for the nal. As usual, as you attempt to solve
any of
Math 113 Homework 2
Graham White
April 20, 2013
Book problems:
1. Consider an arbitrary element v of V . The set cfw_v1 , v2 , . . . vn spans V , so there are scalars a1 , a2 , . . . , an with
v = a1 v1 + a2 v2 + . . . + an vn .
Rearranging this equation
Math 113 Homework 3
Graham White
April 30, 2013
Book problems:
4. To show that V = null(T ) cfw_au, a F, we need to show that V = null(T ) + cfw_au, a F and that
null(T ) cfw_au, a F = cfw_0.
Let v be any element of V . Then v = (v
T(
T (v )
T (u) u)
+
T
Math 113 Final Exam: Solutions
Thursday, June 11, 2013, 3.30 - 6.30pm.
1. (25 points total) Let P2 (R) denote the real vector space of polynomials of degree
2. Consider the following inner product on P2 (R):
1
1
p, q :=
2
p(x)q (x)dx
1
(a) (10 points) U
Math 113 Homework 4
Graham White
May 5, 2013
Book problems:
3. The claim is true. Let U be any subspace of V other than cfw_0 and V itself. Then U has a basis cfw_u1 , u2 , . . . , um .
We have that m 1, because U = cfw_0. Extend this to a basis of V ,
cf
Math 113 Homework 5
Graham White
May 11, 2013
Book problems:
3. For the sake of a contradiction, assume that the set cfw_v, T v, T 2 v, . . . , T m1 v is linearly dependent. Then for
some integer k with 0 k m 1 there is an equation
ak T k v = ak+1 T k+1
Math 113 Homework 1
Graham White
April 20, 2013
Book problems:
3. Let v be an arbitrary element of the vector space V . By the denition of the additive inverse, we have that
(v ) + (v ) = 0
and that
v + (v ) = 0.
Therefore,
(v ) + (v ) = v + (v )
(equalit
MIDTERM SOLUTION
Notation: We use N = cfw_0, 1, 2, . . . to denote the set of all positive integers.
1. Solution: It follows from corollary 3.5 that if there is an injective linear map
V W , then dim(V ) dim(W ).
We then prove suthe converse. Let (e1 , .
Math 113 Midterm ExamSolutions
Held Thursday, May 7, 2013, 7 - 9 pm.
1. (10 points) Let V be a vector space over F and T : V V be a linear operator.
Suppose that there is a non-zero vector v V such that T 3 v = T v. Show that at
least one of the numbers 0
Math 113 Homework 6
Graham White
May 21, 2013
Book problems, Chapter 8:
22. Let cfw_v1 , v2 , v3 , v4 be a basis of C4 . Dene a linear operator T by T (v1 ) = T (v2 ) = 0, T (v3 ) = v3 , T (v4 ) =
v3 + v4 . (Why is there a linear operator taking these va
Math 113 Homework 8
Graham White
May 31, 2013
Book problems
1. From the previous homework set, we know that an orthonormal basis for P2 (R) with the given inner product is
cfw_1, 3(2x 1), 5(6x2 6x + 1). The matrix of T with respect to this basis is
0
3 3
Math 113 Homework 7
Graham White
May 28, 2013
Book problems
10. The rst element of our basis is the function 1.
The second element is proportional to x x, 1 1 = x 1 . Normalising, we get
2
1
12(x 2 ).
The third element is proportional to
1
1
1
1 1
1
x2 x2