Dimension:
Motivation
Dimension
We use the word dimension in everyday speech.
Intuitively, we understand what 2D and 3D mean.
In this lecture, we are going to define dimension
mathematically!
And we wont stop at 3D. you will say with confidence
That space

Midterm Recap
Many answers said: if C is not invertible then BC is not
invertible.
This is false!
1
0
0
1
=
1
0
2
3
1 0
0 0 4
0 15
1 0
0 0
Singular
What we said in lecture is if C is singular then BC is
not invertible.
Recall that singular means square an

Determinants
Reading: Strang 5.1
Determinant
The determinant is a function that takes as input a
square matrix and returns a real number.
This is a single number summary of the matrix.
The determinant of a matrix is nonzero if and only if
the matrix is in

Nullspace
Reading: Strang 3.2
Review: Column Space
Let A be a m-by-n matrix.
Last week we talked about the column space of A :
C(A) = span(cfw_A(:, 1), . . . , A(:, n)) R
m
the set of all linear combinations of the columns of A .
The column space is close

Transpose
Reading: Strang 2.7
Transpose
The transpose of a m-by-n matrix A is a n-by-m matrix
T
denoted A and defined as
AT (i, j) = A(j, i)
AT is the matrix A reflected about its diagonal.
2
1
A=4 6
7
5
4
3
3
2
8 5
1
2
A =4
T
1
5
2
6
4
8
The columns of A

MH1200 Quiz 3
October 13, 2016
Problem 1. Compute the determinant of the following matrix:
a
0
0
0
b
c
0
0
d + g e + h f + i j .
g
h
i
j
Problem 2. Compute the determinant of the following matrix:
0 a 0 0
0 0 b 0
0 0 0 c
d 0 0 0

MH1200 Quiz 3
October 13, 2016
Problem 1. Compute the determinant of the following matrix:
a 0 0 0
b c 0 0
g h i j .
d e f 0
Problem 2. Let
Compute the determinant of A.
a 0 0
f e d
A = b c 0 0 c b .
d e f
0 0 a

MH1200 Quiz 1
September 1, 2016
Problem 1. Let ~a = (a1 , a2 ). Find a vector ~b = (b1 , b2 ) such that h~a, ~bi = 0. (Express the
components of ~b in terms of those of ~a.)
Problem 2. Let ~b = (b1 , b2 , b3 ) and
2 1 0
A = 1 2 1 .
0 1 2
Compute A~b.

MH1200 Quiz 2
September 15, 2016
Problem 1. Let
1 1 1
A = 0 0 3 .
0 2 4
Give an elementary matrix E such that EA is upper triangular.
Problem 2. Determine the values of b for which the system of linear equations
1
1 1 1
x1
2 4 6 x2 = b
2
2 0 2 x3
has n

MH1200 Quiz 1
September 1, 2016
Problem 1. Find two vectors ~u, ~v perpendicular to (1, 0, 1) and to each other.
Problem 2. Let ~b = (b1 , b2 , b3 ) and
1 0 0
A = 1 1 0 .
1 1 1
Compute A~b.

MH1200 Quiz 5
November 10, 2016
Problem 1. A matrix A and its reduced row echelon form rref(A) are given as follows
1 2 1 2 0
1 2 0 1 1
1 2 1 2 0
0 0 1 1 1
A=
rref(A)
=
1 2 1 0 2
0 0 0 0 0 .
1 2 1 0 2
0 0 0 0 0
1. Give a basis for the row space of A.
2. G

MH1200 Quiz 2
September 15, 2016
Problem 1. Let
1 1 1
A = 0 1 3 .
0 2 4
Give an elementary matrix E such that EA is upper triangular.
Problem 2. Determine the values of b for which the system of linear equations
1
1 1 1
x1
1 3 5 x2 = b
2
2 0 2 x3
has a

MH1200 Quiz 4
October 27, 2016
Problem 1. Is it possible to construct a 3-by-3 matrix whose column space contains the vectors
(1, 1, 1) and (1, 0, 1) but does not contain the vector (5, 2, 1)? Justify your answer.
Problem 2. Let M3,3 be the set of all 3-b

MH1200 Quiz 5
November 10, 2016
Problem 1. A matrix A and its reduced row echelon form rref(A) are given as follows
1
2 4 5 3
1 0 0 5 1
1
0 1 0 5 1
2 4 5 3
A=
rref(A)
=
1 2 1 0 1
0 0 1 5 0 .
3 2 1 0 1
0 0 0 0 0
1. Give a basis for the row space of A.
2. G

MH1200: Linear
Algebra I
Course Info
Lectures: Tue 14:30-15:30, Wed 8:30-10:30, LT27
Tutorials: Thu beginning Aug 18.
Instructor: Troy Lee
Office hours: Tue, Thu 16:00-17:00, SPMS-MAS-05-02
Contact me: troyjlee@gmail.com
Grading
Quizzes: 20% of your grad

Row Swaps
Reading: Strang 2.3
Elementary Matrix
A matrix that implements a row operation of
Gaussian elimination is called an elementary matrix.
We have seen the form of elementary matrices
corresponding to adding a multiple of one row to another.
2
1
40

Determinants
Reading: Strang 5.2
Big Formula
Now we derive the big formula for the determinant.
With the cofactor formula, we used linearity to expand
the determinant along one row.
The big formula doesnt stop here, but keeps expanding
along each row succ

Dimension
Reading: Strang 3.5
Review: Basis
Let V be a vector space. A basis for V is a sequence
of vectors v1 , . . . , vn such that
1) span(cfw_v1 , . . . , vn ) = V
2) the sequence v1 , . . . , vn is linearly independent.
By (1), we can write every ele

Matrices: Quick
Preview
Reading: Strang 1.3
Matrix
A matrix is a rectangular array of numbers.
They pop up all over the place.
A classy matrix: MH1200
SG
MY
CN
ID
VN
Other
M+E
58
8
10
0
1
1
Math
126
13
18
5
4
8
Phys
14
0
1
2
1
0
Other
6
0
8
0
3
4
Basic Te

Matrix Multiplication
Reading: Strang 2.3, 2.4
Overview
Today we learn how to multiply matrices together.
We motivate matrix multiplication through Gaussian
elimination.
We can formulate the row operations of Gaussian
elimination in terms of the actions o

Storytime
Carl Friedrich Gauss
30 April 1777 - 23 February 1855
A German mathematician who had a
tremendous impact on many areas of
mathematics and physics.
But he did not discover Gaussian elimination
This was already a standard technique for solving lin

Inverse by cofactors
Reading: Strang 5.3
Review: Cofactors
Let A be an n-by-n matrix.
0
0
(i,
j)
The cofactor of the
entry is det(Aij ), where Aij is
equal to A outside of the ith row and in the ith row
is zero everywhere except for the (i, j) entry, whic

Vector Operations
Reading: Strang 1.1
Properties of vector
addition
Commutativity
~u + ~v = ~v + ~u
Associativity
(~u + ~v ) + w
~ = ~u + (~v + w)
~
This means it doesnt matter in what order we add
the vectors up.
Distributivity Properties
Scalar multi

Vector Spaces: Elementary
Consequences
Review: Vector Space
A vector space is a nonempty set V on which the
operations of addition and scalar multiplication
are defined and satisfy the 10 conditions C1-C2, A1-A4,
M1-M4.
C1) Closure under addition:
x+y 2V

Dimensions of the
Four Subspaces
Reading: Strang 3.6
Dimensions of the Four
Subspaces
Let A be an m-by- n matrix.
Say after Gaussian elimination A has r pivots.
nullspace N (A) R
dimension: n
n
dimension: r
N (A ) R
dimension: m
r
row space C(A ) R
T
left

Announcement
Midterm is next week Tuesday 14:30-15:30 in Exam
Hall C.
Please take note of your tutorial
number for the midterm
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
T11
T12
08.30-09.30
08.30-09.30
08.30-09.30
08.30-09.30
09.30-10.30
09.30-10.30
09.30-10.30
09.30

Inverses
Reading: Strang 2.5
Computing the right inverse
Now we understand when the right inverse exists and
that we can compute it by solving n systems of linear
equations.
Now we will see a way to organize the computation
of the inverse.
We solve all n

Gaussian Elimination
Reading: Strang 2.2
Overview
Last week we began talking about linear equations.
We developed some geometric intuition for what the
solution set to a system of linear equations looks like.
Now we get down to the business of computing t

Inverses
Reading: Strang 2.5
Definition: Invertible
Definition: A square matrix A is invertible if and only
if there is a matrix B such that
AB = I
and
BA = I
Here I is the identity matrix of the same size as A .
A square matrix that is not invertible is

MH1200 Quiz 4
October 27, 2016
Problem 1. Is it possible to construct a 3-by-3 matrix whose column space contains the vectors
(1, 1, 0) and (1, 0, 1) but does not contain the vector (1, 1, 1)? Justify your answer.
Problem 2. Let M3,3 be the set of all 3-b