THE UNIVERSITY OF MANITOBA
DATE: October 22, 2012
Midterm Examination
DEPARTMENT & COURSE NO. MATH 1300
PAGE NO: 1 of 6
EXAMINATION: Vector Geometry & Linear Algebra
TIME: 1 Hour
Always show (justify) your work unless otherwise stated!
(8)
1. Solve, by Ga
1
B07.
MATH 1300: Test #3 (Fall 2010)
Solution & marking scheme:
k
0
0
2
[7] 1. Find all values of k for which the matrix A = 1 k 2k
0
7
10
k+3
Solution. The matrix is not invertible when det(A) = 0 . Computing:
det(A) = (k + 1)(k 2 2k)(k + 3) and this
1
B14.
MATH 1300: Test #2
Solutions
1 0 0
1. Use row reduction to find the inverse of the matrix 0 1 1 . (No marks will be given if other
1 1 2
methods are used.)
Solution. Start with the augmented matrix then row reduce up to RRE form:
1 0 0 1 0 0
1
1
MATH 1300: Test #1
Solutions
B13.
1. Use Gauss-Jordan elimination to solve the following system. Show your work describing
your steps. State clearly your final answer. (No marks will be given if you do not use Gauss-Jordan
elimination!)
x + y z = 0
2y 2
1
MATH 1300: Quiz #4
Solutions
B13.
1. (a) Consider the set S of all polynomials of type ax 2 + (2a 3)x + b , where a and b
range through the set of all real numbers. Is S a subspace of the vector space IP2 of all polynomials
of degree at most 2? Justify
1
B15.
MATH 1300: Quiz #3
Solutions
1. Suppose u = (1, 2, 3) and v = (2,1, 0)
(a) Compute 2u v .
(b) Find the components of a vector w such that 2w v = u .
(c) Find the unit vector in the direction of v.
(d) Find any (non-zero) vector z that is perpendicu
1
MATH 1300: Quiz #4
Solutions
B16.
1. Given u = (1, 2, 3) and v = (2,1, 0) compute u v .
Solution.
1 3 1 2
2 3
uv =
,
,
= (3, 6, 5) .
1 0 2 0 2 1
2. Find the parametric equations of the line passing through the point P(1, 2, 3) and
orthogonal to the pl
1
B05.
MATH 1300: Test #2 (Fall 2010)
Solution & marking scheme:
1 1 2
[8] 1. Use row reduction to find the inverse of the matrix 0 0 1 (You will get no points if
0 1 2
other methods are used.)
Solution.
1 1 2 | 1 0 0
1 1 2 | 1 0 0
1 0 0 | 1 0 1