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The midterm exam for the math 232 linear algebra course offered at simon fraser university in spring 2012. The exam consists of 5 questions, including matrix operations, orthogonal sets, true/false questions related to linear algebra, solving a linear system, and finding the equation of a plane. The exam is 59 points in total and lasts for 50 minutes.
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MATH 232 D100 Spring 2012 Instructor: D. J. Katz February 1, 2012, 11:30 a.m. – 12:20 p.m.
Name: (please print) family name given name
SFU ID: @sfu.ca student number SFU-email
Signature:
Instructions:
√ (^2).
Question Maximum Score
, u =
, and v =
[2] (a) Compute AB.
[2] (b) Compute BT^.
[2] (c) Compute A−^1.
[2] (d) Compute uvT^.
you get 3 points for the right answer, 0 points for the wrong one, and 1. 5 points for leaving it blank. You do not need to justify your answers.
[3] (a) If A is a square matrix whose columns form an orthonormal set, then A must be invertible.
True False
[3] (b) If A and B are square matrices of the same size, then tr(AB) = tr(A) tr(B).
True False
[3] (c) A consistent linear system with two equations in three unknowns can not have only one solution.
True False
[3] (d) If A is a square matrix, and the linear systems Ax = 0 and Ax = b have different numbers of solutions, then A must not be invertible.
True False
[3] (e) A matrix with linearly independent columns must have linearly independent rows.
True False
[3] (f) If k is a positive integer and the span of S = {v 1 , v 2 ,... , vk} is the same as the span of T = {v 1 , v 2 ,... , vk− 1 }, then S must be linearly dependent.
True False
2 − 2 3 a + 2 1 a 1 1 − 1 4
x y z
a
Determine the value(s) of a, if any, for which this system has no solutions. Determine the value(s) of a, if any, for which this system has precisely one solution. Determine the value(s) of a, if any, for which this system has infinitely many solutions.