7.7
Projections
P. Danziger
1
Components and Projections
e
e
e
e
ee v
u
e
projv u
Given two vectors u and v, we can ask how far we will go in the direction of v when we travel
along u. The distance we travel in the direction of v, while traversing
Ryerson University
Lab 3  MTH 141
1. Problem A5 (b), Section 1.3 Determine whether the following vectors are
orthogonal to each other
2
3
1 , 1
7
1/ 3
2. Problem A7 (a), Section 1.3 Find a scalar equation of the plane that
contains the given point wi
Ryerson University
Lab 5  MTH 141
1. Problem A2 (c) Section 3.1 Calculate the following product
2
1
5
3
1
3
4
4
3
0
2
3
1
2
2
3 1
2. Problem A7 (b) Section 3.1 Let A = 3 1 4 and consider Ax
1
0 1
as a
linear
combination of columns of A to determine x
Ryerson University
Lab 8  MTH 141
1. Problem A1 (d) Section 3.5 For the following matrix, either show that the
matrix is not invertible or nd its inverse.
0 0 1
0 1 1
1 1 1
1 0 1
2. Problem A2 (b) Section 3.5 Let B = 2 1 3 Use B 1 to nd the
1 0 2
1
sol
Ryerson University
Lab 12  MTH 141
1. Problem A3 (d) Section 5.1 Evaluate the determinants of the following
matrix by expanding along the row or column of your choice
4 0
1
1 0 6
1 0 3
2 3
6
3
4
1
4
2. Problem A1 (c) Section 5.2 Use row operations an
Ryerson University
Lab 10  MTH 141
1. Problem A1 (b) Section 4.4 Check that the given vectors in B are linearly
independent(and therefore form a basis for the subspace they span). then
determine the coordinates of x and y with respect to B.
B = cfw_1 + x
Ryerson University
Lab 9  MTH 141
1. Problem A1 (e) Section 4.2 Determine, with proof, if the following set is
a subspace of the given vector space.
a1
a3
a2
a4
a1 a4 a2 a3 = 0, a1 , a2 , a3 , a4 R
of M (2, 2)
2. Problem A2 (a) Section 4.2 Determine, wi
Ryerson University
Lab 4  MTH 141
1. Problem A5(c) Section 2.1 For the following system of linear equations
i) Write the augmented matrix
ii) Obtain a row equivalent matrix in row echelon form
iii) Determine whether the system is consistent or inconsiste
Ryerson University
Lab 10  MTH 141
1. Problem A1 (c) Section 4.5 Prove that the following mapping is linear
a b
c d
tr : M (2, 2) R dened by tr
=a+d
2. Problem A3 (b) Section 4.5 Determine whether the given vector y is in
the range of the given linear ma
1
MTH 141 Test 1
DEPARTMENT OF MATHEMATICS
MIDTERM TEST #1
MTH 141 LINEAR ALGEBRA
. First Name:
Last Name (Print):
. Student Number:
Section (circle one)
Dr. Alvarez :
Signature:
.
Date: Oct 11, 2013, 4:00 pm
Duration: 1.5 hours
1 2 3 4 5
Dr. Liu :
6 7 8
1
MTH 141 Final Exam
DEPARTMENT OF MATHEMATICS
FINAL EXAM
MTH 141 LINEAR ALGEBRA
. First Name:
Last Name (Print):
. Student Number:
Section (circle one)
Dr. Alvarez :
Signature:
.
Date: Dec 4th, 2013, 7:00 pm
1 2 3 4 5
Dr. Liu :
6 7 8 9 10
Dr. Jung :
11 1
Ryerson University
Lab 2  MTH 141
1. Problem A7 (c), Section 1.1 Write a vector equation of the line passing
4
through P (2, 0, 5) with the direction d = 2
11
2. Problem A8 (d), Section 1.1 Write a vector equation for the line that
passes through the po
Ryerson University
Lab 1  MTH 141
1. Problem A2(b) Section 9.1 Express the following product in standard
form: (2 4i)(3 i)
2. Problem A3(a) Section 9.1 Determine the complex conjugate of 3 5i
3. Problem A5(c) Section 9.1 Express the following quotient
Ryerson University
Lab 6  MTH 141
1. Problem A1 (c) Section 3.3 Determine the matrix of the rotation in the
plane through the angle = /4.
2. Problem A2 Section 3.3.
a) In the plane, what is the matrix of a stretch S by a factor of 5 in the
x2 direction?
1
Vectors in Rn
P. Danziger
1
Vectors
The standard geometric denition of vector is as something which has direction and magnitude but
not position.
Since vectors have no position we may place them wherever is convenient.
Vectors are often used in Physics

Systems of Equations
P. Danziger
1
Systems of Equations
We can have more than one equation which we want to be simultaneously true.
1.0.1
2 Equations in 2 Unknowns
a1 x + b 1 y = c 1
a2 x + b 2 y = c 2
Here a1 , a2 , b1 , b2 , c1 and c2 are all constant
1.3
Lines and Planes in R3
P. Danziger
1
Lines in R3
We wish to represent lines in R3 . Note that a line may be described in two dierent ways:
By specifying two points on the line.
By specifying one point on the line and a vector parallel to it.
If we a

Trig Review
P. Danziger
1
1.1
Trigonometric Review
Radians
In this course we use radians to measure angles.
The circumference of a circle radius r is 2r.
So in traversing a unit circle we travel a distance 2.
If we traverse half way we travel a distance

Introduction
P. Danziger
1
Linear Algebra
Linear
Algebra
of line or line like
Manipulation, Solution or
Transformation
Thus Linear Algebra is about the Manipulation, Solution and Transformation of line like objects.
We will also investigate the spaces
3.2, 3.3
Inverting Matrices
P. Danziger
1
Properties of Transpose
Transpose has higher precedence than multiplication and addition, so
AB T = A B T
and A + B T = A + B T
As opposed to the bracketed expressions
(AB)T and (A + B)T
Example 1
1 2 1
Let A =
an
1.2
Gaussian Elimination
P. Danziger
1
m Equations in n Unknowns
Given n variables x1 , x2 , . . . , xn and n + 1 constants a1 , a2 , . . . , an , b the equation
a1 x1 + a2 x2 + . . . + an xn = b
represents an n 1 dimensional object in nspace, called a h
3.1
Matrices
P. Danziger
1
Matrices
1.1
Denitions
Denition 1
1. A Matrix is an m n (m by n) array of numbers.
a11 a12 . . . a1n
a21 a22 . . . a2n
.
.
.
.
.
.
.
.
.
.
.
am1 am2 . . . amn
2. The entries in a matrix are called the components of the matrix
2.2
Homogeneous Equations
P. Danziger
Theorem 1 Given a system of m equations in n unknowns, let B be the m (n + 1) augmented
matrix. Recall r is the number of leading ones in the REF of B, also the number of parameters in
a solution is n r.
If r = n, th
Ryerson University
Lab 7  MTH 141
1. Problem A1 (b) Section 3.4 Let L be the linear mapping with matrix
1
0
1
2
0
1
2
5
1
3
3
and let y2 = 5
1
1
1
5
Is y2 in the range of L? If so, nd x such that L(x) = y2 .
2. Problem A2 (a) Section 3.4 Find a ba
1
MTH 141 Test 2
DEPARTMENT OF MATHEMATICS
MIDTERM TEST #2
MTH 141 LINEAR ALGEBRA
. First Name:
Last Name (Print):
. Student Number:
Section (circle one)
Dr. Alvarez :
Signature:
.
Date: Nov 15, 2013, 4:00 pm
1 2 3 4 5
Dr. Liu :
6 7 8 9 10
Dr. Jung :
11 1