Difference between revisions of "09-240/Classnotes for Tuesday September 22"

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(Class notes for today: Add vector section before examples.)
(Proof of VS4: Incomplete examples, and food for thought)
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Set <math>y = \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix}</math> and note
 
Set <math>y = \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix}</math> and note
 
: <math>x + y = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} + \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix} = \begin{pmatrix} a_1 + (-a_1) \\ a_2 + (-a_2) \\ \vdots \\ a_n + (-a_n) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix} = 0_{F^n}</math>
 
: <math>x + y = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} + \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix} = \begin{pmatrix} a_1 + (-a_1) \\ a_2 + (-a_2) \\ \vdots \\ a_n + (-a_n) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix} = 0_{F^n}</math>
 +
 +
=== Examples ===
 +
 +
# <math>F^n \mbox{ for } n \in \mathbb N</math>
 +
#: <math>F^n = \left\{ \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} : a_i \in F \right\}</math>
 +
#: <math>\begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \\ \vdots \\ a_n + b_n \end{pmatrix}</math>
 +
#: <math>a \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix} = \begin{pmatrix} ab_1 \\ ab_2 \\ \vdots \\ ab_n \end{pmatrix}</math>
 +
#: ...
 +
# <math>\mathrm M_{m \times n}(F)</math>
 +
#: ...
 +
 +
=== Food for thought ===
 +
 +
What is wrong with setting
 +
 +
<math>
 +
\begin{pmatrix}
 +
2 & 3 \\
 +
4 & 5 \\
 +
\end{pmatrix} \cdot \begin{pmatrix}
 +
6 & 7 \\
 +
8 & 9 \\
 +
\end{pmatrix} = \begin{pmatrix}
 +
2 \cdot 6 & 3 \cdot 7 \\
 +
4 \cdot 8 & 5 \cdot 9 \\
 +
\end{pmatrix} = \begin{pmatrix}
 +
12 & 21 \\
 +
32 & 45 \\
 +
\end{pmatrix} ?
 +
</math>
 +
 +
# Unnecessary for a V.S.
 +
# This is useless

Revision as of 17:46, 22 September 2009

Contents

Some links


WARNING: The notes below, written for students and by students, are provided "as is", with absolutely no warranty. They can not be assumed to be complete, correct, reliable or relevant. If you don't like them, don't read them. It is a bad idea to stop taking your own notes thinking that these notes can be a total replacement - there's nothing like one's own handwriting! Visit this pages' history tab to see who added what and when.

Class notes for today

Vectors:

  1. can be added
  2. can be multiplied by a number (not another vector)

Let \mathcal F be a field. A vector space \mathbf V over the field \mathcal F is a set \mathbf V (of vectors) with a special element 0_V, a binary operation + : \mathbf V \times \mathbf V \rightarrow \mathbf V, a binary operation \cdot : \mathcal F \times \mathbf V \rightarrow \mathbf V.

Convention for today:
x, y, z \in \mathbf V
a, b, c \in \mathcal F

VS1 \forall x, y \in \mathbf V, x + y = y + x
VS2 \cdots (x + y) + z = x + (y + z)
VS3 \cdots x + 0 = x
VS4 \forall x, \exists y \mbox{ s.t. } x + y = 0
VS5 1 \cdot x = x
VS6 a \cdot (b \cdot x) = (a \cdot b) \cdot x
VS7 a \cdot (x + y) = ax + ay
VS8 (a + b) \cdot x = ax + bx

Proof of VS4

Take an arbitrary x = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} \in F^n

Set y = \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix} and note

x + y = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} + \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix} = \begin{pmatrix} a_1 + (-a_1) \\ a_2 + (-a_2) \\ \vdots \\ a_n + (-a_n) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix} = 0_{F^n}

Examples

  1. F^n \mbox{ for } n \in \mathbb N
    F^n = \left\{ \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} : a_i \in F \right\}
    \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \\ \vdots \\ a_n + b_n \end{pmatrix}
    a \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix} = \begin{pmatrix} ab_1 \\ ab_2 \\ \vdots \\ ab_n \end{pmatrix}
    ...
  2. \mathrm M_{m \times n}(F)
    ...

Food for thought

What is wrong with setting


\begin{pmatrix}
2 & 3 \\
4 & 5 \\
\end{pmatrix} \cdot \begin{pmatrix}
6 & 7 \\
8 & 9 \\
\end{pmatrix} = \begin{pmatrix}
2 \cdot 6 & 3 \cdot 7 \\
4 \cdot 8 & 5 \cdot 9 \\
\end{pmatrix} = \begin{pmatrix}
12 & 21 \\
32 & 45 \\
\end{pmatrix} ?

  1. Unnecessary for a V.S.
  2. This is useless