Math 115A/5, Fall 2012
Linear Algebra
Homework assignment 1, due Friday October 5th
Sections 1.2 and 1.3 Vector spaces and subspaces
1. Read and understand all of the examples in Sections 1.2 and 1.3.
115A-6, Syllabus and General informations
Fall 2012
Prerequisite: Math 33A
Text: Linear Algebra, by Friedberg, Insel, and Spence, with a supplement on Languages and Proofs,
and Induction (custom editi
Course 115A - Midterm1
Y. Tendero
October 19th, 2012
Name:
ID Number:
Signature:
Instructions: Show all work to receive full credit. Feel free to use the back of each
paper but please indicate that yo
1
Homework 1 (Due Oct 5th)
Proof is the heart of mathematics. It distinguishes mathematics from the sciences and
other disciplines. Courts of law deal with the burden of proof, juries having to decide
1
Homework 5 (Due Nov 2nd)
Ex1: Do Ex5 section 2.2
Ex2: Do Ex1 section 2.3 a) and e) to j)
Ex3: Do Ex 4c) section 2.3
Ex4: Do Ex 10 of section 2.3
Ex5: Do Ex 1 of section 2.4
Ex6: Do Ex 2 of section 2
Course 115A - Midterm2
Y. Tendero
November 19th, 2012
Name:
ID Number:
Signature:
Instructions: Show all work to receive full credit. Feel free to use the back of each
paper but please indicate that y
1
Homework 9 (Due Fri, Nov 30th)
Ex1: Let A be a n n matrix with n distinct eigenvalues 1 , , n . Show that
det(A) = 1 n and tr(A) = 1 + + n .
1
1
Ex2: Find a 2 2 matrix A which has v1 =
and v2 =
has
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certai
1
Homework 3 (Due Oct 19th)
Ex1:
Prove:
Lemma 1.1. (span(S) is bigger than S)
Let S be a subset of a vector space V then S span(S).
Ex2:
Prove the second part: (Any subspace containing S also contains
1
Homework 10 (Due Fri, Dec 7th)
Ex1: Do Ex 2 section 6.1.
Ex2: Do Ex 3 section 6.1.
Ex3: Do Ex 4 section 6.1.
Ex4: Do Ex 5 section 6.1.
Ex5: Do Ex 8 section 6.1.
Ex6: Do Ex 9 section 6.1.
Ex7: Do Ex
1
Homework 6 (Due Nov 9th)
Ex1: Let = (1, 0), (0, 1) be the standard basis for R2 and let = (3, 4), (4, 3)
be another basis for R2 . Let l be the line connecting the origin to (4, 3) and T : R2 R2
be
QUIZ 1 (MATH 61, FALL 2012)
Your Name:
UCLA id:
Math 61 Section:
Date:
The rules:
This is a multiple choice quiz. You must circle exactly one answer with an ink pen.
If two or more answers are circled
QUIZ 2 (MATH 61, FALL 2012)
Your Name:
UCLA id:
Math 61 Section:
Date:
The rules:
This is a multiple choice quiz. You must circle exactly one answer with an ink pen.
If two or more answers are circled
Math 115A/5, Fall 2012
Linear Algebra
Homework assignment 1, due Friday October 5th
Sections 1.2 and 1.3 Vector spaces and subspaces
2. Let V be the set of real 3-tuples:
V = cfw_ (a1 , a2 , a3 ) : a1
Math 115A/5, Fall 2012
Linear Algebra
Homework assignment 2, due Monday October 15th
Section 1.4 Linear combinations
1. Let V be a vector space and let W1 and W2 be subspaces of V . Dene the subset
W1
1
Homework 4 (Due Oct 26th)
Ex1:
Lemma 1.1. Let T : V W be a linear transformation. The null space N (T ) is a
subspace.
Proof. Write the proof.
Ex2:
Lemma 1.2. (R(T ) is a subspace)
Let T : V W be li
1
Homework 2 (Due Oct 12th)
For exercises, keep in mind that weve a double goal: linear algebra and proofs. Each
answer must be carefully justied, step by step. No argument can be skipped. When one
ch
1
Homework 7 (Due Nov 16th)
Ex1: Let U, V, W be nite-dimensional vector space, and let S : V W and T : U V
be linear transformations.
1. Show that rank(ST ) rank(S)
2. Show that rank(ST ) rank(T )
3.
1
Homework 8 (Due Nov 26th)
Ex1: Let A and B be similar n n matrices. Show that A and B have the same set of
eigenvalues (every eigenvalue of A is an eigenvalue of B and vice versa).
Ex2: For this que
Let F be a eld, and S = cfw_s1 , s2 , s3 be a set with exactly three elements. Consider
the vector space F(S, F ) all functions from S to F with the standard function addition
and scalar multiplicati