SAMPLE PROBLEMS FOR 353-Midterm 2
TRUE FALSE
PROBLEMS 1 FROM CHAPTERS 4.1, 4.2, 4.3, 4.4, 5.1, 5.2, 5.3.
1. For any nn matrix A, and scalar k F , we have det(kA) = k det(A).
2. det(AAT A1 ) = detA.
3. A square matrix A is invertible if detAk = 0 for any n
MA 3530000160150
MA 3530000260151
GOINS
HOMEWORK ASSIGNMENT #6
Due Friday, September 30, 2016 at the start of lecture. This should be turned in at the
start of class, and should be legible and stapled. Late assignments will not be accepted.
(The problems
MA 3530000160150
MA 3530000260151
LESSON 6: FRIDAY, SEPTEMBER 2, 2016
GOINS
1.6: Bases and Dimension
Recap.
Theorem 1.8. Let V, +, be a vector space over a field F . Given a finite
subset = cfw_u1 , u2 , . . . , un V, the following are equivalent:
i. We
MA 3530000160150
MA 3530000260151
LESSON 3: FRIDAY, AUGUST 26, 2016
GOINS
1.4: Linear Combinations and Systems of Linear Equations
Recap. We showed the following result in the previous lecture.
Theorem 1.3. Assume that V, +, is a vector space over a field
MA 3530000160150
MA 3530000260151
LESSON 2: WEDNESDAY, AUGUST 24, 2016
GOINS
1.3: Subspaces
Definitions. Let F denote a field ; the only examples which will interest us are
Q, the collection of rational numbers p/q;
F = R, the collection of real numbers a
MA 3530000160150
MA 3530000260151
GOINS
HOMEWORK ASSIGNMENT #2
Due Friday, September 2, 2016 at the start of lecture. This should be turned in at the
start of class, and should be legible and stapled. Late assignments will not be accepted.
(The problems a
MA 3530000160150
MA 3530000260151
LESSON 4: MONDAY, AUGUST 29, 2016
GOINS
1.5: Linear Dependence and Linear Independence
Recap. Let V, +, be a vector space over a field F , and choose a finite subset S = cfw_u1 , u2 , . . . , un
of V. For any scalars a1
MA353 SPRING 2012 Homework 5
This rst problem will be a lot of work. We did something similar in lecture on March 7 and
9, so you probably want your notes from those lectures handy. Its a great problem because
it draws on several aspects of the course fro
MA353 SPRING 2012 Homework 7:Sylvesters Criterion and
Recurrence Relations
Denition 1. Let A be a matrix/linear operator. We write A > 0 if A is positive and
invertible, that is Ax, x > 0 for all nonzero vectors x.
1. For each x R dene the matrix
5
2 1
Ax
MA 3530000160150
MA 3530000260151
GOINS
HOMEWORK ASSIGNMENT #1
Due Friday, August 26, 2016 at the start of lecture. This should be turned in at the
start of class, and should be legible and stapled. Late assignments will not be accepted.
(The problems app
MA 3530000160150
MA 3530000260151
LESSON 5: WEDNESDAY, AUGUST 31, 2016
GOINS
1.6: Bases and Dimension
Recap. Let V, +, be a vector space over a field F , and choose a subset S = cfw_u1 , u2 , . . . , un
of V. We say that S is a linearly dependent set if
MA 3530000160150
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LESSON 1: MONDAY, AUGUST 22, 2016
GOINS
1.1: Introduction
2-Dimensional Vectors. Any entity involving both magnitude and direction is called a vector.
For instance, say that we consider a point P = (0, 0) to be the origin
Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence
Jephian Lin, Shia Su, Zazastone Lai July 27, 2011
Copyright 2011 Chin-Hung Lin. Permission is granted to copy, distribute and/or modify this document un
SAMPLE PROBLEMS FOR 353 nal
TRUE FALSE
.
1. Let W1 , W2 , and W3 be subspaces of a vector space V , then (W1 +
W2 ) W3 is a subspace of V .
2. If dim(V ) = n then any set of n-vectors is linearly independent if and
only if it is linearly independent.
3. L
Math 54 Discussion Section SOLUTIONS
Rob Bayer
April 14, 2008
Dont freak out. Some of these problems are meant to be hard.
1. Prove that if A is invertible, then AB is similar to BA
AB = A(BA)A1 , so we can just use P = A in the denition of similar.
2. Le
SAMPLE PROBLEMS FOR 353-Midterm 1
TRUE FALSE
1. Let W1 , W2 , ., Wr be subspaces of a vector space V , then W1 . . .Wr
is a subspace of V .
2. Consider S1 S2 V two sets of vectors of a vector space V . If S2 is
linearly dependent, then so is S1 .
3. Let V
MATH353-001\002
Final Exam Solution
December 9th, 2013
Instructor: CHING-JUI LAI
1
2
1. True and false. You dont need to explain the reason. Note
that a vector space V can be innite dimensional if not specied.
(4 points each, 80 points in total.)
T Let W1
Fall 2012
Exam #3 Key
Math 306
1. Denitions and Short Answer: (3 Points Each) Give a denition or short answer for each
of the following.
(a) A Subspace of a vector space V . A subset H of a vector space V is a subspace of V
if 0 H and for all vectors x an
MATH353-001\002
Final Exam
December 9th, 2013
Instructor: CHING-JUI LAI
NAME:
PUID:
There are in total 200 points in this exam.
You can use both sides of the paper.
No calculator.
Mark clearly your nal answer.
1
8
2
9
3
10
4
11
5
12
6
13
7
Total
1
2
1
SUMMARY OF MA 35100: ELEMENTARY LINEAR ALGEBRA
Course Description
The following can be found here: https:/www.math.purdue.edu/academic/courses/MA35100/
Credit Hours: 3.00. Systems of linear equations, finite dimensional vector spaces,
matrices, determinan
MA353 SPRING 2012 Homework 6/Notes
1. Theorems from lecture
Recall the following items from lecture: Let T : V V be a linear operator (as always, a
nite-dimensional inner product space, either real or complex).
Denition 1. We dene the norm of T as
T = max
Purdue University MA 353: Linear Algebra II with Applications Homework 11, due Apr. 9, Some solutions Sec. 5.4:#16: Let T be a linear operator on a finite-dimensional vector space V . (a) Prove that if the characteristic polynomial of T splits, then so do
Purdue University MA 353: Linear Algebra II with Applications Homework 10, due Apr. 2, Some Solutions (I) Sec. 1.6#34: (a) Prove that if W1 is any subspace of a finite-dimensional vector space V , then there exists a subspace W2 of V such that V = W1 W2 .
Purdue University MA 353: Linear Algebra II with Applications Homework 9, due Mar. 26, Some Solutions (I) Sec. 5.1:#17 Let T be the linear operator on Mnn (R) defined by T (A) = At . (a) Show that 1 is the only eigenvalues of T . Proof. Let be an eigenval
Purdue University MA 353: Linear Algebra II with Applications Homework 8, due Mar. 12, Some Solutions (I) Sec. 5.1#8 (a) Prove that a linear operator T on a finite-dimensional vector space is invertible if and only if zero is not an eigenvalue of T . Proo
Purdue University MA 353: Linear Algebra II with Applications Homework 7, due Mar. 5, Some Solutions sec. 2.4 #4: Let A and B be n n invertible matrices. Prove that AB is invertible and (AB)-1 = B -1 A-1 . Proof. Recall that AB is invertible iff there exi
Purdue University MA 353: Linear Algebra II with Applications Homework 6, due Feb. 26, Some Solutions Sec. 2.2 #13: Let V and W be vector spaces, and let T and U be nonzero linear transformations from V to W . If R(T ) R(U ) = cfw_0, prove that cfw_T, U
Purdue University MA 353: Linear Algebra II with Applications Homework 5, due Feb. 19, Solutions Sec. 2.1 #9 Prove that there exists a linear transformation T : R2 R3 such that T (1, 1) = (1, 0, 2) and T (2, 3) = (1, -1, 4). What is T (8, 11). Proof. Note