Linear Algebra MATH 2050U
TERM TEST 2
Instructor: M. Beligan
Fri. March 22, 2013
Name (last name, rst name):
Student Number:
Teaching Assistant (circle one): Christine Dabrowski
Luis Zarrabeitia
Instructions
Before starting, read over the entire test car

OBJECTIVES:
a Section 1.5 By the end of this section, you will be able:
to state how you get an elementary matrix from the identity
to state that row reduction can be performed by multiplication of the appropriate elementary
matrix
to state that if A i

OBJECTIVES:
a Section 6.3 By the end of this section, you will be able:
to state that a set of vectors is an orthogonal set if all pairs of vectors in the set are orthog-
onal
to state that an orthogonal set of vectors in which all vectors have norm 1 i

OBJECTIVES:
o Sections 3.4-3.5 By the end of these sections you will be able:
to determine whether or not two lines are parallel, or whether or not two lines are orthogonal
to determine whether or not two planes, or a line and a plane are parallel, or w

All matrices are nxn unless otherwise specified.
T F
T F
T F
T F
T F
T F
trace(AB) = trace(BA)
T F
T F
If A is upper triangular, then adj(A) is too.
T F
T F
The product of two elementary matrices is
an elementary matrix.
T F
T F
If Ax=b is a system of lin

MATH 2050U Term Test 1 Page 2 of 6
1. [6 marks] Solve the linear systems associated with the following matrices:
J 3- xs xu 36: X\ X2. X3
1 2 0 3 0 2 1 2 3 1
(i) 0 0 1 1 0 2 ] (ii) [ 0 1 2 1 ]
0 0 0 U 1 4 0 U 1 1
NUAeELs
THE 'X 1: _.
Ger lxg=ul 1 k 9"
5

OBJECTIVES:
a Section 4.8 By the end of this section, you will be able:
to nd the rank of a matrix
to nd the nullity of a matrix
to connect rank and nullity of a matrix with notions of invertibility, and of solving systems of
Hnearequaons
HONOUR HOMEWO

OBJECTIVES:
a Section 2.1 By the end of this section, you will be able:
to write down any minor or cofactor of any matrix.
to compute the determinant of a 3 x 3 matrix from cofactor expansion along either rows or
columns
to write down the matrix of cof

OBJECTIVES:
a Section 1.3 By the end of this section, you will be able:
to determine the size of a matrix
to determine whether two matrices are equal
to write down examples of column vectors and row vectors
to determine from the sizes of the matrices

OBJECTIVES:
a Section 1.2. By the end of this section, you will be able:
to recognize when a matrix is in reduced rowechelon form
to recognize when a matrix is in rowechelon form
to determine from an augmented matrix in row-echelon or reduced rowechelo

OBJECTIVES:
0 Section 4.2 By the end of this section, you will be able:
— to give the deﬁnition of a ”subspace of a vector space" , and determine whether or not a given
subset of a vector space is a subspace
— to determine whether or not a vector, w, is a

OBJECTIVES:
a Section 5.2 By the end of this section, you will be able:
to write down that a matrix A is diagonalizable if there is an (invertible) matrix P such that
D = PlAP is a diagonal matrix
to nd the matrix P that diagonalizes a 3 x 3 matrix A
t

OBJECTIVES:
. Section 5.1 By the end of this section, you will be able:
to write down the equation that denes the eigenvalues and eigenvectors of a matrix
to state that if x is an eigenvector of a matrix A with a real eigenvalue t, then the vector Ax
is

OBJECTIVES:
a Section 2.2 By the end of this section, you will be able:
to determine the determinant of a matrix (of any size) with a row of zeros or a column of
zeros.
to determine the determinant of a matrix (of any size) given the determinant of its

OBJECTIVES:
a Section 4.1 By the end of this section, you will be able:
to use the vector space axioms to determine whether or not any given set of objects (with a
rule for vector addition and scalar multiplication) constitutes a vector space
to state t