Math 25a Homework 10
Due Wednesday, November 29, 2006
The problem set is divided into three parts. Please put the rst part in
Scotts box, the second in Gerardos, and the third in Matts. Remember to
staple each bundle of solutions and also to put your name
Lecture 4
Now we have one of two main subspace theorems. It says we can extend a basis for a
subspace to a basis for the full space.
Theorem 0.1 (One subspace theorem). Let W be a subspace of a nite-dimensional vector
space V . If BW is a basis for W , th
Lecture 1
We should start with administrative stu from the syllabus. Instructor, grader, undergrad
tutors, solicitation for oce hour times, text, homework due on Tuesday in class. Grade
breakdown.
Linear algebra is the study of linear functions. In Rn the
Lecture 3: Steinitz exchange and bases
Recall the intuition that a set is linearly independent if each vector in it is truly needed
to represent vectors in the span. Not only are the all needed, but linear independence implies
that there is exactly one wa
Lecture 5
We saw last time that we can add linear transformations and multiply them by scalars.
These are just two ways to generate new linear transformations. Another obvious one is
composition.
Proposition 0.1. Let T : V W and U : W Z be linear (with al
Lecture 2
Examples.
1. For all n1 n2 , Cn1 is a subspace of Cn2 (as C-vector spaces).
2. Given a vector space V over F, cfw_0 is a subspace.
3. In R2 , any subspace is either (a) R2 , (b) cfw_0 or (c) a line through the origin. Why?
If W is a subspace and
Lecture 6
We can give an alternative characterization of one-to-one and onto:
Proposition 0.1. Let T : V W be linear.
1. T is injective if and only if it maps linearly independent sets of V to linearly independent sets of W .
2. T is surjective if and onl
Lecture 7
Last time we saw that if V and W have dimension n and m and we x bases B of V and
C of W then there is an isomorphism : L(V, W ) Mm,n (F) given by
(T ) = [T ]B .
C
A simple corollary of this follows. Because of any basis is a basis, these spaces
Lecture 10
When we looked at the dual V and constructed the dual basis B for a basis B, the dual
element v to an element v B actually depended on the initial choice of basis B. This is
because to dene v , we must express a vector in terms of coordinates u
Lecture 9
Dual maps
Given T : V W that is linear, we will dene a corresponding transformation T t on the
dual spaces, but it will act in the other direction. We will have T t : W V .
Denition 0.1. If T : V W is linear, we dene the function T t : W V by th
Math 25a Homework 1
Due Wednesday, September 27, 2006
1
Sets and Maps
(1) Let A, B X . Prove that A B if and only if X B X A.
(Hint: This is an if and only if statement, so you have to prove both that the rst statement
implies the second and that the seco
Math 25a Homework 6
Due Wednesday, November 1, 2006
The problem set is divided into three parts. Please put the rst part in
Scotts box, the second in Gerardos, and the third in Matts. Remember to
staple each bundle of solutions and also to put your name o
Math 25a Homework 5
Due Wednesday, October 25, 2006
The problem set is divided into three parts. Please put the rst part in
Scotts box, the second in Gerardos, and the third in Matts. Remember to
staple each bundle of solutions and also to put your name o
Math 25a Homework 9
Due Wed, Nov 22 or Mon, Nov 27 (if away for Thanksgiving)
The problem set is divided into three parts. Please put the rst part in
Scotts box, the second in Gerardos, and the third in Matts. Remember to
staple each bundle of solutions a
Math 25a Homework 11
Due Wednesday, December 6, 2006
The problem set is divided into three parts. Please put the rst part in
Scotts box, the second in Gerardos, and the third in Matts. Remember to
staple each bundle of solutions and also to put your name
Math 25a Homework 12
Due Wednesday, December 13, 2006
The problem set is divided into three parts. Please put the rst part in
Scotts box, the second in Gerardos, and the third in Matts. Remember to
staple each bundle of solutions and also to put your name
Math 25a Homework 13
Due Tuesday, December 19, 2006
The problem set is divided into three parts. Please put the rst part in
Scotts box, the second in Gerardos, and the third in Matts. Remember to
staple each bundle of solutions and also to put your name o
Math 25a Homework 8
Due Wednesday, November 15, 2006.
The problem set is divided into three parts. Please put the rst part in
Scotts box, the second in Gerardos, and the third in Matts. Remember to
staple each bundle of solutions and also to put your name
Math 25a Homework 7
Due Thursday, November 9, 2006.
The problem set is divided into three parts. Please put the rst part in
Scotts box, the second in Gerardos, and the third in Matts. Remember to
staple each bundle of solutions and also to put your name o
Math 25a Homework 3
Due Wednesday, October 11, 2006.
The problems are divided into three batches. Each of our CAs will grade one batch,
so please put your name on and staple each batch of problems separately. Many thanks!
1
First batch of problems
(1) (a)
Math 25a Homework 2
Due Wednesday, October 4, 2006
1
Injections, surjections, bijections
(1) Prove that the composition of two injective functions is an injective function and that
the composition of two surjective functions is a surjective function.
(2)
Math 25a Homework 4
Due Wednesday, October 18, 2006
The homework set is divided in three batches. Each of our CAs will grade one batch.
Remember to staple each batch of solutions separately and also to put your name on each!
For some of the questions in t
Lecture 8: Dual spaces
We have been talking about coordinates, so lets examine them more closely. Let V be an
n-dimensional vector space and x a basis B = cfw_v1 , . . . , vn of V . We can write any vector
V in coordinates relative to B as
a1
[v]B = , wh