Math 136 - January 4 2016 - Lecture 1

Prof: Dan Wolczuk
Office Hours: T -> 11:30 - 12:20 am , 1:30 - 3:30 pm ; Th -> 11:30 - 12:20 am ; 3:00 - 5:30 pm
Office:
Email:
Text Book: Linear Algebra Course Notes 3.0 by Wolczuk. Edition 2.1 is fine.
Grades: Final Exam: 65% ; Midterm: 25% ; Assignments: 10%
Tests: Midterm: Monday Feb 8th, 7:00 - 8:50 pm ; Final: TBA
Assignments are due Wednesdays by 3:00pm in MC 4066

Introduction

Objectives for this course:

• To give you a solid understanding of many of the fundamental concepts in linear algebra
• To show you the importance of understanding definitions, theorems, and proofs
• To improve your ability to read and write proofs

30% of the marks on the tests will be for proofs, the rest are for computations.

Vectors in $\mathbb{R}^n$

You should be familiar with 2 and 3 dimensional Euclidean spaces, often denoted $\mathbb{R}^2$ and $\mathbb{R}^3$ . $\mathbb{R}^2$ is the set of all points of the form (x,y) where x, y $\epsilon \ \mathbb{R}$ and $\mathbb{R}^3$ is the set of all points (x, y, z), where x, y, z $\epsilon \ \mathbb{R}$

We will now view them as a set of vectors.

Definition:
$$\begin{equation} \{ \mathbb{R}^n = \begin{bmatrix} x_1\\ \vdots \\ x_n \end{bmatrix} | x_1 , ... , x_n \epsilon \ \mathbb{R} \} \end{equation}$$

For example, the vector $\vec{x}$ =
$$\begin{bmatrix} 1 \\ 2 \end{bmatrix}$$ can be thought of as the point (1,2)
Remark: All vectors in this course start at the origin

The set of all vectors of the form $\vec{x}$ = $$\begin{bmatrix} s \\ t \\ 0\end{bmatrix}$$ , s ,t $\epsilon \ \mathbb{R}^3$ can be thought of as the set of all points of form (s,t,0). This represents the xy plane in $\mathbb{R}^3$

Definition $$\begin{equation} \begin{aligned} Let \ \vec{x} & = \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix} \\ \ \vec{y} & = \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix} \\ \end{aligned} \end{equation}$$
We say that $\vec{x}$ and $\vec{y}$ are equal and write $\vec{x} = \vec{y}$ if $x_i = y_i$ for all 1 $\leq$ i $\leq$ n

We define addition by $\vec{x} + \vec{y} =$ $$\begin{bmatrix} x_1 + y_1 \\ \vdots \\ x_n + y_n\end{bmatrix}$$
for any real scalar $t \ \epsilon \ \mathbb{R}$ we define scalar multiplication by
$t\vec{x} =$ $$\begin{bmatrix} tx_1 \\ \vdots \\ tx_n \end{bmatrix}$$

We need to define equality first as it is used in to define addition and scalar multiplication.

We can only add vectors with the same dimensions.

Definition
Let $\vec{v}_1 ...\vec{v}_k$ be vectors in $\mathbb{R}^n$, we call a sum of scalar multples of their vectors a linear combination. That is, linear combinations of $\vec{v}_1 ... \vec{v}_k$ is:
$$\begin{equation} t_1 \vec{v}_1 + t_2\vec{v}_2 + ... + t_k\vec{v}_k \end{equation}$$ where $t_1, t_2 ... t_k$ are real constants

Theorem 1.1.1

If x , y , w $\epsilon \ \mathbb{R}^n$ and c,d $\epsilon \ \mathbb{R}$. then :
V1 $\vec{x} + \vec{y} \ \epsilon \ \vec{R}^n$
V2 $(\vec{x} + \vec{y}) + \vec{w} = \vec{x} + (\vec{y} + \vec{w})$
V3 $\vec{x} + \vec{y} = \vec{y} + \vec{x}$
V4 $\exists \ \vec{0} \ \epsilon \ \mathbb{R}^n$ | $\vec{x} + \vec{0} = \vec{x} \ \forall \ \vec{x} \ \epsilon \ \mathbb{R}^n$
V5 $\forall$ x in $\mathbb{R}^n \ \exists$ a vector ($\vec{-x}$) $\exists \ \mathbb{R}^n$ | $\vec{x} + \vec{-x} = \vec{0}$
V6 c$\vec{x} \ \exists \ \mathbb{R}^n$
V7 c(d $\vec{x}$) = cd($\vec{x}$)
V8 (c+d)$\vec{x}$ = c$\vec{x}$ + d$\vec{x}$
v9 c($\vec{x}$ + $\vec{y}$) = c$\vec{x}$ + c$\vec{y}$
v10 1$\vec{x}$ = $\vec{x}$

Math 136 - Jan 6 2016 - Lecture 2

Spanning

There are 3 kinds of mathematicians, those who can count, and those who can't

In linear algebra, sets of all possible linear combinations of a set of vectors $\mathcal{B} = \{ \vec{v_1} ... \vec{v_k} \}$ in $\mathbb{R}^n$ is called the span of the set $\mathcal{B}$ and we write:

S $=$ Span $\mathcal{B} =$ Span $\{ \vec{v_1} ... \vec{v_k} \}$
$= \{t_1\vec{v_1} + ... + t_k\vec{v_k}\}$

We also say that S is spanned by $\mathcal{B}$ and that $\mathcal{B}$ is a spanning set for S.

We will sometimes want to write a set differently. For a set:
$$\begin{equation} \{ t_1\vec{v_1} + ... + t_k\vec{v_k} | t_1 , .. , t_k \epsilon \mathbb{R} \} \end{equation}$$

We will just write:
$$\begin{equation} \vec{x} = \{ t_1\vec{v_1} + ... + t_k\vec{v_k} + \vec{b} \} , \ t_1, ... , t_k \epsilon \ \mathbb{R} \end{equation}$$
and call this a vector equation of the set.

Notice that the vector equation gives us the general form of any vector in the set

Insert a whole bunch of examples here. Aka read the textbook for those.