MA 265 GOINS SAMPLE MIDTERM EXAMINATION #1
Name:
Instructions:
Circle the correct answer on the following pages. You have 50 minutes to complete 35 problems. No textbooks, personal notes, calculators, or computing aids are allowed during the examination p

MA 265 LECTURE NOTES: WEDNESDAY, FEBRUARY 20
Subspaces Denition. Let (V, , ) be a real vector space. We say that a subset W V is a subspace of V if (1) W = , and (2) (W, , ) is a real vector space. We spend the rest of the lecture giving examples and disc

MA 265 REVIEW FOR MIDTERM #1
GOINS
1: Linear Equations and Matrices 1.1: Systems of Linear Equations a system of m linear equations in n unknowns is in the form a11 x1 a21 x1 . . . am1 x1 + + a12 x2 a22 x2 . . . + + + + + a1n xn a2n xn . . . = = b1 b2 . .

MA 265 GOINS MIDTERM EXAMINATION #1
Name:
Instructions:
Circle the correct answer on the following pages. You have 50 minutes to complete 35 problems. No textbooks, personal notes, calculators, or computing aids are allowed during the examination period.

MATH 265 Exam 1, 09/23/2004
1. Consider the follow vectors in R4 . 1 1 -1 1 v1 = , v2 = , 1 1 -1 1
1 1 v3 = , -1 -1
6 2 w = , 1 -3
Determine whether w is in spancfw_v1 , v2 , v3 . If your answer is affirmative, give also the coefficients for expressing

MATH 265 Second Midterm Examination7 November 13, 2007
Copyright 2007, Clarence Wilkerson
All rights reserved
PUID :
No books, notes, or calculators allowed. Please be neat and show alllyour work on these
pages (use back of pages if necessar

Math 265, Practice Midterm 1
Sept 19, 2012
Name:
This exam consists of 8 pages including this front page.
Ground Rules
1. No calculator is allowed.
2. Show your work for every problem unless otherwise stated.
3. You may use one 4-by-6 index card, both sid

MA 265 Midterm I
PRACTICE
NAME:
Student ID:
Books, notes or any kind of calculator are NOT allowed.
Problem Score
1&2
3
4
5
6
Bonus
Total
Honor Pledge: I have neither received nor given aid on this exam.
Signature:
1
c
d
a b
1. (5 pts) Let A = x ex cos x

Math 265
Professor Priyam Patel
11/2/13
Midterm 2 Practice Problems
You should look over the Midterm 2 Topics Review, the multiple choice problems I listed
on the website and also use these problems as practice. Homework problems and handout
problems are

NAME:
Math 265
Midterm Exam 1
October 1, 2013
Instructions:
(1) Show your work. No credit will be given for unsupported answers to problems requiring
computation. You may receive credit for partially correct work even if your nal answer is
incorrect.
(2)

Practice problems1
1. How many dierent 2 2 matrices are in reduced row-echelon forms?
A.
1
B.
2
C.
3
D.
4
E.
Innitely many.
2. If A, B T are both 2 3, then what is the size of AB?
A.
23
B.
22
C.
33
D.
32
E.
None of the above
3. If A2 = A, what can we say

Practice problems1
1. Quick questions:
A.
What are the requirements for a set of vectors in a vector space V to form a basis
of V ?
B.
What are the minimum and maximum possible ranks of an m n matrix? What
are the minimum and maximum possible dimensions o

MATH 351 EXAM 1 REVIEW
1. Outline of what we have covered roughly Sections 1.1, 1.2 and 1.3 of
Penneys Linear Algebra and the concept of Column Space in Section 1.4
Matrices: size of matrix, adding matrices (when this is dened), scalar multiple of
matrix

MA 265 GOINS MIDTERM EXAMINATION #2
Name:
Instructions:
Circle the correct answer on the following pages. You have 50 minutes to complete 25 problems. No textbooks, personal notes, calculators, or computing aids are allowed during the examination period.

MA 265 GOINS SAMPLE MIDTERM EXAMINATION #2
Name:
Instructions:
Circle the correct answer on the following pages. You have 50 minutes to complete 25 problems. No textbooks, personal notes, calculators, or computing aids are allowed during the examination p

MA 265 REVIEW FOR MIDTERM #2
GOINS
4: Real Vector Spaces 4.1: Vectors in the Plane and 3-Space A vector in the plane or a 2-vector is a 2 1 matrix x = x y . The real numbers x and y are called
the components of x. Given two ordered pairs P (x, y ) and Q(x

MA 265 LECTURE NOTES: MONDAY, FEBRUARY 18
Real Vector Spaces Properties of n-Space. In the previous lecture, we studied properties of V= R2 R3 as 2-space; and as 3-space.
If u, v, w V and c, d R then the following properties are valid: (Commutativity: ) u

MA 265 LECTURE NOTES: WEDNESDAY, FEBRUARY 13
Real Vector Spaces Vectors in n-Space. Recall that Rn is the collection of all real n-vectors: x1 x2 x= . . . xn
where the entries xi are real numbers. We call this collection (real) n-space. The elements x Rn

MA 265 LECTURE NOTES: MONDAY, FEBRUARY 11
Cramers Rule Let A be an n n matrix. We have seen that if det(A) = 0, then A is nonsingular. In fact, its inverse is the n n matrix A1 = Aij 1 adj(A) = det(A) det(A) a c b d
T
=
Aji . det(A)
We give a brief exampl

MA 265 LECTURE NOTES: FRIDAY, FEBRUARY 8
Computing Determinants Determinants of Special Matrices. Let A be an n n matrix. (1) Assume that A is upper triangular: a11 a12 a1n a22 a2n A= . . . . . . ann
Then det(A) = a11 a22 ann is the product of the element

MA 265 LECTURE NOTES: WEDNESDAY, FEBRUARY 6
Properties of Determinants (contd) Type II: Multiplying Rows and Columns. Let A be an n n matrix. We explain how det(A) changes if we apply an elementary operation of Type II. (1) (2) (3) (4) Let B be that matri

MA 265 LECTURE NOTES: MONDAY, FEBRUARY 4
Determinants Review. In the previous lecture, we dened the determinant of an n n matrix as the following number: a11 a21 . . . an1 a12 a22 . . . an2 . . a1n a2n . . . ann
=
Sn
( ) a1 (1) a2 (2) an (n)
as a sum ove

MA 265 LECTURE NOTES: FRIDAY, FEBRUARY 1
Finding Inverses of Matrices (contd) Computing Inverses. In the last lecture, we showed that an n n matrix A is invertible if and only if A is row equivalent to the n n identity matrix In . In other words, say that

MA 265 LECTURE NOTES: WEDNESDAY, JANUARY 30
Finding Inverses of Matrices (contd) Nonsingularity. Let E = E1 E2 Ek1 Ek be a product of elementary matrices. Then E is nonsingular. We began the proof of this statement in the previous lecture. It suces to sho