## Math 1600 Lecture 2, Section 2, 8 Sept 2014

### Announcements:

Read Section 1.2 for next class. Work through homework problems.

Lecture notes (this page) available from course web page. Also look for announcements there.

No tutorials this week. There is a quiz in tutorials next week.

Please read over syllabus, especially before e-mailing me with questions, as it covers all of the main points.

Let me know if the bookstore runs out of texts or combo packs.

### Review of last lecture:

A vector can be represented by its list of components, e.g. $[1, 2, -1]$ is a vector in $\R^3$.
We write $\R^n$ for the set of all vectors with $n$ real components, e.g. $[1, 2, 3, 4, 5, 6, 7]$ is in $\R^7$.

We also often write vectors as column vectors, e.g. $\coll 1 2$.

Vector addition: $[u_1, \ldots, u_n] + [v_1, \ldots, v_n] := [u_1 + v_1, \ldots, u_n + v_n]$.
E.g. $[3, 2, 1] + [1, 0, -1] = [4, 2, 0]$.

Scalar multiplication: $c [u_1, \ldots, u_n] := [c u_1, \ldots, c u_n]$.

E.g. $2 [ 1 , 2, 3, 4, 5] = [2, 4, 6, 8, 10]$.

Zero vector: $\vec{0} := [0, 0, ..., 0]$.

### New material: Section 1.1, continued: Properties of vector operations

The picture to the right shows geometrically that vector addition is commutative: $\vec{u} + \vec{v} = \vec{v} + \vec{u}$.

In this true in $\R^n$? Let's check: \begin{aligned} \vec{u} + \vec{v}\ \ &= [u_1 + v_1,\, \ldots,\, u_n + v_n] \\ &= [v_1 + u_1,\, \ldots,\, v_n + u_n] \\ &= \vec{v} + \vec{u}. \end{aligned}

Many other properties that hold for real numbers also hold for vectors: Theorem 1.1. But we'll see differences later.

Example: Simplification of an expression: \begin{aligned} &3 \vec{b} + 2 (\vec{a} - 4 \vec{b})\\ =\ &3 \vec{b} + 2 \vec{a} - 8 \vec{b}\\ =\ &2 \vec{a} - 5 \vec{b} \end{aligned}

True/false: For every vector $\vu$, we have $2 \vu = \vu + \vu$.

True/false: For every vector $\vu$, we have $2 \vu \neq 3 \vu$.

### Linear combinations

Definition: A vector $\vv$ is a linear combination of vectors $\vv_1, \vv_2, \ldots, \vv_k$ if there are scalars $c_1, c_2, \ldots, c_k$ so that $\vv = c_1 \vv_1 + \cdots + c_k \vv_k .$ The numbers $c_1, \ldots, c_k$ are called the coefficients. They are not necessarily unique.

Example: Is $\coll 1 {-1}$ a linear combination of $\coll 1 1$, $\coll 2 {-1}$ and $\coll 0 1$?

Yes, since $\coll 1 {-1} = 1 \coll 1 1 + 0 \coll 2 {-1} - 2 \coll 0 1 \qqtext{(Check!)}$

Note: We also have $\coll 1 {-1} = -\frac{1}{3} \coll 1 1 + \frac{2}{3} \coll 2 {-1} + 0 \coll 0 1 \qqtext{(Check!)}$ and many more possibilities.

We will learn later how to find all solutions.

### Coordinates

Example: Express $\vw_1 = \coll 3 3$ as a linear combination of $\vu = \coll 2 1$ and $\vv = \coll {-1} 1$.

We can solve this by using $\vu$ and $\vv$ to make a new coordinate system in the plane. Use the board to show that $\vw_1 = 2 \vu + \vv$.

Similarly, show that $\vw_2 = \coll 4 {-1}$ can be expressed as $\vw_2 = \vu - 2 \vv$.

Note that in this case the coefficients are unique. In this situation, the coefficients are called the coordinates with respect to $\vu$ and $\vv$. So the coordinates of $\vw_1$ with respect to $\vu$ and $\vv$ are $2$ and $1$, and the coordinates of $\vw_2$ with respect to $\vu$ and $\vv$ are $1$ and $-2$.

Working in a different coordinate system is a powerful tool.

### Binary vectors

$\Z_2 := \{ 0, 1 \}$, a set with two elements.

Multiplication is as usual.

Addition: $0 + 0 = 0$, $\ 0 + 1 = 1$, $\ 1 + 0 = 1$, $\ \red{1 + 1 = 0}$.

$\Z_2^n :=$ vectors with $n$ components in $\Z_2$.

E.g. $[0, 1, 1, 0, 1] \in \Z_2^5$.

$[0,1,1] + [1,1,0] = \query{[1,0,1]}$ in $\Z_2^3$.

There are $\query{2^n}$ vectors in $\Z_2^n$.