MATH 351 MIDTERM 1 REVIEW
1. Outline of what we have covered roughly Sections 1.1, 1.2 and 1.3 of
Penneys Linear Algebra
Matrices: size of matrix, adding matrices (when this is dened), scalar multiple of
matrix, transpose of a matrix
Rn as the set of al
Not the second MA35100 Test
Monday 1st April 2013
Do not work on the front of this cover page
There are 8 questions. Answer all the questions.
The questions are not of equal value. The total number of marks is 100.
MA 35100 HOMEWORK ASSIGNMENT #12 SOLUTIONS
Problem 1. pg. 229; prob. 8
Consider a linear transformation L(x) = A x from Rn to Rm , with ker(L) = cfw_0. The pseudoinverse L+ of L is the transformation from Rm to Rn given by
L+ (y ) = the least-squares solu
MA 35100 HOMEWORK ASSIGNMENT #3 SOLUTIONS
Problem 1. pg. 65; prob. 4 Interpret the following linear transformation geometrically: T (x) = 1 1 -1 1 x.
Solution: Looking at Theorem 2.2.4 on page 63 of the text, this matrix is in the form a -b b a where a=1
MA 351 Spring 2014 HW8 Solutions
16 9 8
14. XA = C X = CB; so X = 6 3 4.
0 0 0
= I as desired.
18. A1 A = I, so A1 is invertible with inverse A. Theorem 3 from this section.
20. B 1 A1 AB = B 1 IB = B 1 B = I. As A, B are invertible, so
MA351: PRACTICE TEST 1
Answer all questions. No calculators or other electronic devices are allowed. For 5 points,
write your name on the question paper and hand it in together with your solutions. For
questions 1 to 4, you may write down the answers with
MA351: PRACTICE TEST 2
Answer all questions. No calculators or other electronic devices are allowed. For questions 1 to
4, you may write down the answers without showing your working.
1. (4 points) Which of the following functions from R3 to R are linear
MA351: EXAM 1 (26.02.2014)
Answer all questions. No books or notes, calculators or other electronic devices are allowed.
For 10 points, write your name on the question paper and hand it in together with your
solutions. For questions 1 to 4, you may write
FOR THE PROBLEMS REQUIRING YOUR WORK, YOU MUST SHOW ALL YOUR WORK TO RECEIVE CREDIT
and WRITE NEATLY. In other words, any problem SAYING SHOW YOUR WORK without any scratch work won't
receive any credit. Be s
Electronic devices must be turned o. Documents are not allowed. For the Exercises part, all
answers must be carefully justied.
Multiple choices, 40 points
1. If A is a n n matrix, such that A2 = A, then 1 and 1 are the only eigenvalues
SAMPLE TEST PROBLEMS FOR MA351
Here we give some past test problems with brief solutions and you are to work out the details.
Problem 1. Consider the following system
3x y +
6x 2y + (2p + 1)z =
9x 3y + (4 p)z = 5 q
For which values o