Vector Spaces
MAT 1341
Spring, 2013
1
1
Examples and Basic Properties
Denition 1.1. A vector space consists of a nonempy set V of elements
(called vectors ) that can be added and multiplied by a number (called a
scalar ), and for which certain properties
Rank-Basis for subspaces
MAT 1341
Spring, 2013
1
1
Basis
Denition 1.1. Let H be a subspace of Rn . A basis of H is a linearly
independent set in H that spans H. In other words,cfw_1 , ., k is a basis
v
v
, ., is linearly independent, and
of H if cfw_ v
Suggested Exercises:
Throughout the semester, there will be weekly updates. You
should check this page regularly. All the exercises are from
Elementary Linear Algebra, Nicholson 2nd Edition.
Week 1:
Suggested Exercises:
Throughout the semester, there will be weekly updates. You
should check this page regularly. All the exercises are from
Elementary Linear Algebra, Nicholson 2nd Edition.
Week 1:
Complex Numbers
MAT 1341
Spring, 2013
1
1
Complex Numbers
1.1
Introduction
We can easily solve the equation x2 4 = 0.
The answer is x = 2; in particular, x is a rational number, even an
integer.
The equation x2 2 = 0 is a bit more tricky.
The solution
Matrix Multiplication
MAT 1341
Spring, 2013
1
1
Matrix Multiplication
1.1
Matrix Multiplication
T
If R = r1 r2 . . . rn is a row matrix and C = c1 c2 . . . cn is a
columns matrix, each with n entries, we dene their dot product to be the
number
r1 c1 + r2
Linear Equations
MAT 1341
Spring, 2013
1
1
Linear Equations
1.1
Linear Equations
Example 1.1. A charity wishes to endow a fund that will provide $50 000
per year for cancer research. The charity has $480 000 and, to reduce risk,
wants to invest in two ban
Matrix Inverses
MAT 1341
Spring, 2013
1
1
Matrix Inverses
1.1
Matrix Inverses
As we have seen, every system of linear equations can be written in
matrix form
AX = B
where the column X is to be determined.
This suggests that we rst investigate how we sol
Matrices
MAT 1341
Spring, 2013
1
1
Matrices
1.1
Matrices
A rectangular array of numbers is called a matrix, and the numbers
themselves are called entries of the matrix.
Matrices are usually denoted by upper case letters A, B , C , etc.
Example 1.1. The
Vector Geometry
MAT 1341
Spring, 2013
1
1
Geometric Vectors
1.1
Coordinate Systems
In the plane, coordinates are introduced as follows:
Choose a point O called the origin,
choose two perpendicular lines through O called the X -axis, and
the Y -axis, an
Diagonalization and Eigenvalues
MAT 1341
Spring, 2013
1
1
1.1
Diagonalization and Eigenvalues
Eigenvalues and Eigenvectors
Let A =
23
,=
v
04
3
2
and =
u
1
. Then:
1
Figure 1: Matrix A acts by stretching the vector x, not changing its direction,
so x is a
Homogeneous Systems
MAT 1341
Spring, 2013
1
1
Homogeneous Systems
1.1
Homogeneous Systems
A system of equations is called homogeneous if all the constant terms
are zero.
Because the constants are all zero, any homogeneous system always has
the trivial s
Determinants
MAT 1341
Spring, 2013
1
1
Cofactor Expansions
1.1
Cofactor Expansions
In a previous section we dened the determinant of a 2 2 matrix
ab
A=
as follows:
cd
det A = det
ab
= ad bc.
cd
We then showed that A has an inverse if det A = 0, and gave
MAT 1341, Spring/Summer 2013 Assignment 3
Due July 18th 10:30 AM.
Late assignments will NOT be accepted. An assignment drop-o box is assigned for this
course and is located at Math department. (KED 585)
Professor: Termeh Kousha
Student Name
Student Number
MAT 1341, Spring/Summer 2013 Assignment 3
Due July 18th 10:30 AM.
Late assignments will NOT be accepted. An assignment drop-o box is assigned for this
course and is located at Math department. (KED 585)
Professor: Termeh Kousha
Student Name
Student Number
MAT 1341, Spring/Summer 2013 Assignment 1
Due May 23rd 8:30 AM.
Late assignments will NOT be accepted. An assignment drop-o box is assigned for this
course and is located at the Math department. (KED 585)
Professor: Termeh Kousha
Student Name
Student Numb
MAT 1341, Spring/Summer 2013 Assignment 2
Due June 20thrd 10:30 AM.
Late assignments will NOT be accepted. An assignment drop-o box is assigned for this
course and is located at Math department. (KED 585)
Professor: Termeh Kousha
Student Name
Student Numb
MAT 1341, Spring/Summer 2013 Assignment 1
Due May 23rd 8:30 AM.
Late assignments will NOT be accepted. An assignment drop-o box is assigned for this
course and is located at Math department. (KED 585)
Professor: Termeh Kousha
Student Name
Student Number
B
MAT 1341, Spring/Summer 2013 Assignment 2
Due June 20th 10:30 AM.
Late assignments will NOT be accepted. An assignment drop-o box is assigned for this
course and is located at Math department. (KED 585)
Professor: Termeh Kousha
Student Name
Student Number