The Univeral Product Code (bar code) on a product is a vector in $\Z_{10}^{12}$,
such as $$\vu = [6,7,1,8,6,0,0,1,3,6,2,4].$$
In Section 1.4, we will learn how error detection works for codes like this.
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.
Office hour: today, 2:30-3:30, MC103B. Drop with any questions!
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.
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: $\vec{0} := [0, 0, ..., 0]$.
Properties of vector operations: The parallelogram shows geometrically that vector addition is commutative: $\vec{u} + \vec{v} = \vec{v} + \vec{u}$.
Many other properties that hold for real numbers also hold for vectors: Theorem 1.1. But we'll see differences later.
An important real-world application:
Derive an equation for Inky's target on board.
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.
Example: Is $\coll 1 {-1}$ a linear combination of $\coll 1 0$ and $\coll 2 0$?
Example: Is $\coll 0 0$ a linear combination of $\coll 1 0$ and $\coll 2 0$?
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.
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 $\Z_2^n$.
To add and multiply, always take the remainder modulo $3$ at the end.
E.g. $2 + 2 = 4 = 1 \cdot 3 + \red{1}$, so $2 + 2 = 1 \pmod{3}$.
We write $\!\!\pmod{3}$ to indicate we are working in $\Z_3$.
Similarly, $1 + 2 = \query{0} \pmod{3}$ and $2 \cdot 2 = \query{1} \pmod{3}$.
$\Z_3^n := $ vectors with $n$ components in $\Z_3$.
$[0,1,2] + [1,2,2] = \query{[1,0,1]}$ in $\Z_3^3$.
There are $\query{3^n}$ vectors in $\Z_3^n$.
E.g., in $\Z_{10}$, $\quad 8 \cdot 8 = 64 = 4 \pmod{10}$.
$\Z_m^n := $ vectors with $n$ components in $\Z_m$.
To find solutions to an equation such as $$ 6 x = 6 \pmod{8} $$ you can simply try all possible values of $x$. In this case, $1$ and $5$ both work, and no other value works.
Note that you can not in general divide in $\Z_m$, only add, subtract and multiply. For example, there is no solution to the following equation: $$ 2 x = 1 \pmod{4} $$ But there is a solution to $$ 2 x = 1 \pmod{5}, $$ namely $x = \query{3}$
Question: In $\Z_5$, what is $-2$?
Example 1.40 (UPC Codes):
The Univeral Product Code (bar code) on a product is a vector in $\Z_{10}^{12}$,
such as $$\vu = [6,7,1,8,6,0,0,1,3,6,2,4].$$
In Section 1.4, we will learn how error detection works for codes like this.