MA351
Practice Exam 1
1. Consider the following linear system of equations:
x + 2y + 3z = 4
2x + z = 3
4y + 7z = 11
(a) Write the linear system into the form AX = B. Write down your A, X, B explicitly.
(b) Form the augmented matrix and reduce it to RREF.
Linear Algebra
Math 351, Linear Algebra
3
Contents
Chapter 1. Module 1: Vector Spaces
1. Lecture 1. The real vector space R2
2. Lecture 2. The real vector spaces Rn
3. Lecture 3. Real vector spaces
4. Lecture 4. Product spaces and projection maps.
5. Lect
Math 351 Fall 2015, PSet 3
1
Problem Set 3: Math 351, Fall 2015
Due Tuesday September 29
September 21, 2015
Do the problems below. Please write neatly, especially your name! Show all your work and justify all your steps.
Write in complete, coherent senten
Math 351 Fall 2015, PSet 4
1
Problem Set 4: Math 351, Fall 2015
Due Tuesday October 1
September 22, 2015
Do the problems below. Please write neatly, especially your name! Show all your work and justify all your steps.
Write in complete, coherent sentences
Math 351 Fall 2015, PSet 6
1
Problem Set 5: Math 351, Fall 2015
Due Thursday November 5
October 29, 2015
Do the problems below. Please write neatly, especially your name! Show all your work and justify all your steps.
Write in complete, coherent sentences
Math 351 Fall 2015, PSet 5
1
Problem Set 5: Math 351, Fall 2015
Due Thursday October 29
October 27, 2015
Do the problems below. Please write neatly, especially your name! Show all your work and justify all your steps.
Write in complete, coherent sentences
Math 351 Fall 2015, PSet 2 Notes
1
Problem Set 2 Notes: Math 351, Fall 2015
September 20, 2015
I will discuss problems 1-3 on problem set 2.
Recall that the direct product of two real vector spaces V1 ,V2 is the real vector space V1 V2 of pairs (v1 , v2 )
Mid-term 1 Review sheet: Math 351 Fall 2015
1
Mid-term 1 Review sheet: Math 351 Fall 2015
September 30, 2015
1
The Exam: Structure and Content
The rst midterm will cover the material in the lecture notes from Lecture 1 (The real vector space R2 ) through
Dependence, Independence, and Spanning: Math 351 Fall 2015
1
Dependence, Independence, and Spanning: Math 351 Fall
2015
October 2, 2015
Dependence and Independence.
Given a real vector space V and a set of vectors B = cfw_v1 , . . . , vn , we have functio
Math 351 Fall 2015, PSet 1
1
Problem Set 1: Math 351, Fall 2015
Due Thursday September 10
September 9, 2015
Do the problems below. Please write neatly, especially your name! Show all your work and justify all your steps.
Write in complete, coherent senten
Special Note: Math 351, Fall 2015
1
Special Note: Math 351, Fall 2015
October 7, 2015
I have decided that it may be in the best interest of all that I provide some additional notes for the exam. These
notes literally say nothing new. I just will try to sa
Practise Midterm 1: Math 351, Fall 2015
1
Practise Midterm 1: Math 351, Fall 2015
October 2, 2015
Problem 1. (10 Points) State the following denitions:
(a) For a subset B of a vector space V to span.
(b) For a subset B of a vector space V to be linearly d
Math 351 Fall 2015, PSet 2
1
Problem Set 2: Math 351, Fall 2015
Due Thursday September 17
September 9, 2015
Do the problems below. Please write neatly, especially your name! Show all your work and justify all your steps.
Write in complete, coherent senten
Math 351 Fall 2015, Midterm 3 Review
1
Midterm 3 Review
November 30, 2015
The third in-class midterm in December 3. The focus will be on determinants, eigenvalues/eigenvectors, and
Gram-Schmidt. There will be computational problems and also theoretical on
MA351
Practice Exam 2
1. Consider the following matrix:
1
2
2
2
4
4
.
3
6
6
2 4 6
(a) Find a basis for the column space.
(b) Find a basis for the row space.
(c) Find a basis for the nulll space.
(d) Find the rank of this matrix.
1
MA351
Practice Exam 2
2.
MA351: PRACTICE TEST 3
Answer all questions (each worth 4 points). Show your working for any question not split into
(a),(b),(c),(d). No calculators or other electronic devices are allowed.
Notation: For natural numbers m and n,
Mm,n (R) = cfw_m n matrice
Math 351 Fall 2015, PSet 7
1
Problem Set 7: Math 351, Fall 2015
Due Thursday December 3
November 27, 2015
Do the problems below. Please write neatly, especially your name! Show all your work and justify all your steps.
Write in complete, coherent sentence
Midterm 2: Math 351, Fall 2015
1
Midterm 2: Math 351, Fall 2015
November 11, 2015
I will pay anyone with a perfect exam 100 USD.
Problem 1. (10 Points) State the following denitions:
(a) For a subset B of a vector space V to be a basis.
(b) For a linear s
Midterm 2 Data and Solutions: Math 351, Fall 2015
Midterm 2 Data and Solutions: Math 351, Fall 2015
November 17, 2015
1
1.1
Data
1030 Class
The overall statistics on Midterm 2 are as follows:
Total Points
Maximum
Minimum
Average
StDev
78
42.5
17
30.95
6.7