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. Bring any questions
to discussion section or ofce hour
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 edition for UCLA). A link to the errata of the book is provi
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 you have done so. No calculator or any document allowed.
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
whether the case against a defendant has been proven b
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 2.4
Ex7: Do Ex 4 of section 2.4
Ex8: Do Ex 9 of section
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 you have done so. No calculator or any document allowed.
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 eigenvectors, is
1
0
not equal to the identity matrix o
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 certainly know a great deal of mathematics - Calculus, Trigon
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 span(S). ):
Theorem 1.2. (span(S) is a subspace, you c
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 17 section 6.1.
Ex8: Do Ex 2 d,h,i of section 6.2.
Ex9:
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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 the operation of reection through l (so if v R2 then T
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, the answer is not accepted.
You are allowed to use on
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, the answer is not accepted.
You are allowed to use on
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 , a2 , a3 R
[If you dont understand the notation nota
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 + W2 by
W1 + W2 = cfw_ x + y : where x W1 and y W2
so
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 linear, R(T ) is a subspace.
Proof. Write the proof.
Ex3:
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
check a proof he/she does not think: just check that each
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. Show that nullity(ST ) nullity(T )
4. Give an example w
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 question, the eld of scalars will be complex numbers C ins
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 multiplication. Find a basis for F(S, F ) (and prove thats indeed a