Chapter 1
Linear Systems and Gaussian Elimination
Section 1.1
Linear Systems and Their Solutions
Discussion 1.1.1 A line in the xy -plane can be represented algebraically by an equation of the form ax + by = c where a and b are not both zero. An equation

Todays Lecture:
Determinant formula for the inverse matrix
Solution of linear systems by Cramers
rule
Chapter 3: Vectors in n-dimensional space
(Euclidean n-Spaces)
Subsets of Euclidean n-Spaces
Subspaces of Euclidean n-Spaces
Definition 2.5.29
Let A

Previous Lecture:
Determinant of a square matrix
Calculated by cofactor expansion along some row or
column:
det( A) ai1 Ai1 ai 2 Ai 2 ain Ain
a1 j A1 j a2 j A2 j anj Anj
where Aij (1)i+j det(Mi j).
Mi j = (n 1) x (n 1) matrix obtained from A by
deleting t

The midterm exam:
On Wednesday 5 October from 10:50 - 12:10 in
Exam Hall C (North Spine). See info section on
edventure for a map of the location.
Be at Exam Hall C by 10:45am (5 mins early).
Material to be tested: everything covered so far
in the cour

Plan for the rest of the course:
Week 10 (next week): finish Ch. 3, Ch.4
Weeks 11-12: General vector spaces (Ch. 5)
Week 13: Revision
Quizzes in Weeks 11 and 12
Change to the course syllabus:
Linear Transformations is omitted.
Section 3.6 of Ch. 3 al

Previous lecture:
Subspaces of Rn and their properties
V
1.
2.
3.
is a subspace of Rn if
V contains the 0 vector
V is closed under addition.
V is closed under scalar multiplication.
The intersection of subspaces is again a
subspace
The union of two sub

Todays lecture:
Chapter 4
How to find a basis for a vector space
Vector spaces associated with matrices:
column space, row space, nullspace
Rank of a matrix
Dimension Theorem for Matrices
Reminder about basis:
A basis for a vector space V ( Rn or a su

Previous lecture:
Connection between subspaces and
homogeneous linear systems:
Theorem: The solution set of a homogeneous
linear system with n variables is a subspace
of Rn .
Linear combinations of vectors
Linear span: the set of all linear
combination

Previous lecture:
Linear combinations of vectors
Linear span: the set of all linear
combinations of a collection of vectors in Rn
Theorem: Linear spans are subspaces of Rn
Connection between linear spans and linear
systems.
Reminder of the connection be

Todays Lecture:
An application of linear algebra (matrices):
Fibonacci numbers in Nature
Elementary matrices:
Elementary row operations by matrix
multiplication.
Theorems about invertible matrices
How to find inverses using elementary row ops.
1
Fibo

Todays lecture:
Column space, row space and nullspace
of a matrix: An example
More about the nullspace of a matrix
Relation between homogeneous and
inhomogeneous linear systems
Reminder about column space, row space,
nullspace, rank and nullity of a ma

Todays lecture:
Chapter 5
General vector spaces
Vector spaces of functions and matrices
Motivational Example:
Let P2(R) denote the set of all polynomial functions of
degree 2 . I.e. all polynomials of the form
f ( x) a0 a1 x a2 x 2
Can add such polynomi

Todays Lecture:
Determinant of a square matrix
Definition and examples
Properties of the determinant
How the determinant changes when
elementary row operations are
performed on the matrix
1
Determinants
For a square matrix A the determinant det(A)
is

Todays Lecture:
Linear systems involving unknown constants
(continued)
Examples of problems that can be solved via linear
systems
Homogenous systems of linear equations
Matrices: basic definitions and some special types
of matrices
Addition, subtract

Todays Lecture:
More about the inverse of a square matrix
Powers of matrices
1
Reminder about invertible matrices:
Let A be a square matrix of order n.
Then A is said to be invertible if there exists a
square matrix B of order n such that
AB I and BA I.

Chapter 2
Matrices
Section 2.1
Introduction to Matrices
Denition 2.1.1 A matrix (plural matrices ) is a rectangular array of numbers. The numbers in the array are called entries in the matrix. The size of a matrix is given by m n where m is the number of

Chapter 3
Vector Spaces (Euclidean n-Spaces)
Section 3.1 Euclidean n-Spaces
Discussion 3.1.1 (Geometric Vectors) A vector can be represented geometrically as a directed line segment or an arrow; the direction of the arrow species the direction of the vect

Chapter 4
Vector Spaces Associated with Matrices
Section 4.1
Row Spaces and Column Spaces
Discussion 4.1.1 Each m n matrix is naturally associated with three vector spaces, namely, the row space, the column space and the nullspace. These three vector spac

Chapter 5 General Vector Spaces
1
Introduction
Let A be an m n matrix and let b Rm . Then we have a linear system Ax = b (1)
with variables x = (x1 , . . . , xn )T Rn . So far in this course we have developed methods to analyze and solve such systems. In

Todays Lecture:
Determining the number of solutions from the
row-echelon form of the augmented matrix
(continued)
Linear systems involving unknown constants
Remark 1.4.7.1
A linear system is inconsistent, i.e. has no solution, if
and only if a row-echel

Todays Lecture:
Addition, subtraction, and scalar
multiplication for matrices (continued)
Matrix multiplication
1
Reminder
An m x n matrix is an array of numbers with m rows
and n columns:
a11 a12 a1n
a21 a22 a2 n
A
am1 am 2 amn
It is also deno

Todays Lecture:
How to solve systems of linear equations
Solving linear systems whose augmented
matrix is in row-echelon form
Solving linear systems in general by Gaussian
Elimination or Gauss-Jordan Elimination
Determining the number of solutions from

Todays lecture:
More about systems of linear equations
The augmented matrix
Elementary row operations
Row echelon form of the augmented matrix
Reminder: A system of linear equations
is a collection of linear equations in the variables x1,
x2, , xn :
a11

Todays Lecture:
Properties of matrix multiplication
Writing linear systems as matrix
equations
The trace of a square matrix
The transpose of a matrix
The inverse of a matrix
1
Reminder: Matrix multiplication
Let A (aij)m x p and B (bij)p x n be two m

MH1200 Linear Algebra I
Todays lecture:
Overview of the course
Systems of linear equations: basic definitions and
examples
Lines, planes and linear equations
What is linear algebra?
Linear:
Algebra:
Involves lines, planes etc
Straight and flat stuff
So

Todays Lecture:
More about invertibility for square matrices
1
Reminder:
Theorem 2.4.5 (Characterization of Invertibility)
If A is a square matrix, then the following
statements are equivalent:
1. A is invertible.
2. The linear system Ax 0 has only the t

Todays lecture:
Theorems about vector spaces
Examples of finite-dimensional vector
spaces (continued from last lecture)
Subspaces of polynomials and matrices
Theorems about Vector Spaces
Main goals:
Show that every subspace of Rn has a basis.
(Do next