Notes 1: Some Applications of Linear Algebra
What is linear algebra? To begin with it has to do with equations of the form
3x + 2y = 1
x + 4y = 7,
(1)
(2)
that is, equations that do not involve products of variables or trig functions or exponentials
or .

Notes 11a: Markov Processes, Dimension, Rank-Nullity
Lecture Oct , 2011
Application 1. We are given a submersible self-propelled vehicle with three propulsion
devices. Each of these is able to provide thrust in a single direction by any amount,
positive o

Notes 12: Coding
Lecture Oct 22, 2009
We begin with an example of a behavior of matrix multiplication. We can have matrices
A, B so that neither A nor B is the zero matrix, but AB is the zero matrix. This happens
when the image of B is contained in the ke

Notes 13: Coordinates
Lecture Oct , 2011
Assume there are two supermarkets, one called A and the other B in a town of 1000
shoppers and that each month
1. 80% of the shoppers of A remain with A and the rest switch to B, and
2. 70% of the shoppers of B rem

Notes 15: Vector Spaces
Lecture November, 2011
Denition 1. A vector space is a set V with an operation we call addition and a map
that associates to an element R and an element v V another element in V denoted
by v. We denote the addition of two elements

Notes 16: Vector Spaces: Bases, Dimension, Isomorphism
Lecture November, 2011
Let V be a vector space.
Denition 1. Let v1 , v2 , vm V . A linear combination of the elements vi is any
element of V of the form m ai vi , ai R.
1
Denition 2. Let S V . The spa

Notes 17: Bases, Coordinates, Matrices
Lecture November, 2011
Let V, W be vector spaces whose dimension is nite. Let F : V W be a linear
map. We show how to assosciate to F a matrix. We can not do this if we are only given
the map F . We need more informa

Notes 19: Determinants
Lecture November, 2011
Let A be an n n matrix. The determinant of A is a number denoted det(A) or |A|.
It has many uses, but we will use it for one thing. It enables us to decide if a matrix is
invertible.
Theorem 1. Let A be an nn

Linear Algebra and Geometry
We construct a dictionary between some geometrical notions and some notions from
linear algebra.
Adding, Scalar Multiplication
An element of (x, y) R2 corresponds to an arrow with tail at the origin in R2 and head
at the point

Notes 11: Dimension, Rank Nullity theorem
Lecture Oct , 2011
Denition 1. Let V be a subspace of Rn . A basis of V is a subset S of V provided
the set S spans V, and
the set S is independent.
Denition 2. Let V be a subspace of Rn . The dimension of V is

Notes 10: Bases of Kernels, Images: Geometry
Denition 1. As subset S of Rn that is closed under addition and scalar multiplication
is said to be a subspace of Rn . More concretely, a subset S of Rn is a subspace of Rn
provided the two conditions below are

How to Solve Linear Equations
We give an algorithm to nd out if a set of linear equations has a solution, and, if it
does have a solution, how to nd all of the solutions.
Step One: First we rewrite the set of linear equations dropping much of the redundan

Notes 4: Dention of Function, Matrix Multiplication
The material in this section corresponds roughly (only roughly) to section 2.1.
Denition 1. A function consists of two sets, the doman and the target, and a rule which
assigns to every element in the dom

Notes 5: First Properties of Matrices and their Geometry
If two matrices have the same size, then we can add them by adding componentwise.
2 3
1 3
Example 1. Let A = 1 0 , B = 2 4, then
4 7
4 5
1
0
A + B = 3 4 .
0
2
We can multiply a matrix by a scalar.

Notes 6: Geometry of Linear Functions and Multiplication of Matrices
Let v = (1, 2) R2 . Dene the function
projv : R2 R2
to be the orthogonal projection onto the line through v. What is the matrix representing
this function? Note that this is an example o

Notes 7: The Inverse of a Matrix
We show how to solve certain systems of linear equations. The ingredients of this
approach are two. First is the idea of the inverse function to a function. The second idea
is Gauss elimination and the ideas around Gauss e

Notes 7a: A Markov Process
We study the behavior of a xed group of 100 people, so over time the group always
has 100 people. We assume that each person in the group is a member of one of three
states or categories. In our study these states are:
The pers

Notes 8: Kernel, Image, Subspace
Fix a matrix M of size n m.
Question. For what b Rn does the system of equations
Fx = b
have a solution?
We can ask this same question in another way. The matrix M induces a linear map:
M : Rm Rn . By denition the image of

Notes 9: Bases of Kernels, Images
Lecture Oct 6, 2009
Basis of the Kernel
Denition 1. As subset S of Rn that is closed under addition and scalar multiplication is
said to be a subspace of Rn . More concretely S is a subspace provided the two conditions
be

Math 2860, Exam 1 practice problems
September 15, 2015
1. Find the area of the parallelogram formed by the vectors 2 + 3k and 6 + 3k. Ans:
i j
j
405
2. Find the angle between the two vectors in the previous problem. Ans: 96.86 degrees
3. Find all values o