09-240/Classnotes for Thursday September 24 - Revision history

Raymct at 02:47, 22 October 2009

2009-10-22T02:47:23Z

← Older revision		Revision as of 22:47, 21 October 2009
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	# Cancellation: <math>x + y = x + z \Rightarrow y = z</math> (add ''w'' to both sides s.t. ''x'' + ''w'' = 0)		# Cancellation: <math>x + y = x + z \Rightarrow y = z</math> (add ''w'' to both sides s.t. ''x'' + ''w'' = 0)
	# 0 is unique		# 0 is unique
	# Negatives are unique: <math>x + y = 0 = x + z \Rightarrow y = 0</math>		# Negatives are unique: <math>x + y = 0 = x + z \Rightarrow y = z</math>
	# <math>0x = 0. a \cdot 0 = 0</math>		# <math>0x = 0. a \cdot 0 = 0</math>
	# <math>(-a) \cdot x = a \cdot (-x) = -(ax)</math>		# <math>(-a) \cdot x = a \cdot (-x) = -(ax)</math>

Raymct at 02:44, 22 October 2009

2009-10-22T02:44:55Z

← Older revision		Revision as of 22:44, 21 October 2009
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	\|}		\|}

	A V.S. over <math>F: V, 0, +, \~~times~~</math> s.t.		A V.S. over <math>F: V, 0, +, \cdot</math> s.t.

	VS1 <math>\forall x, y \in \mathbf V, x + y = y + x</math><br />		VS1 <math>\forall x, y \in \mathbf V, x + y = y + x</math><br />

C8sd: /* Examples */ Fix indentation, forced png.

2009-10-02T22:25:45Z

Examples: Fix indentation, forced png.

← Older revision		Revision as of 18:25, 2 October 2009
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	# <math>O_{\mathcal F(S, F)}(\sigma) = 0_F \forall \sigma \in S</math>		# <math>O_{\mathcal F(S, F)}(\sigma) = 0_F \forall \sigma \in S</math>
			# <math>f, g \in \mathcal F(S, F)</math><br /><math>\,\! (f + g)(\sigma) = f(\sigma) + g(\sigma)</math><br /><math>f \in \mathcal F(S, F)</math><br /><math>a \in F \Rightarrow \forall \sigma \in S, S(af)(\sigma) = a \cdot (f(\sigma))</math>
	# <math>f, g \in \mathcal F(S, F)</math>
	#: <math>(f + g)(\sigma) = f(\sigma) + g(\sigma)</math>
	#: <math>f \in \mathcal F(S, F)</math>
	#: <math>a \in F \Rightarrow \forall \sigma \in S, S(af)(\sigma) = a \cdot (f(\sigma))</math>

	'''Claim''': + is associative. Given <math>f, g, h \in \mathcal F(S, F), (f + g) + h = f + (g + h) \forall \sigma</math>		'''Claim''': + is associative. Given <math>f, g, h \in \mathcal F(S, F), (f + g) + h = f + (g + h) \forall \sigma</math>
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	<li><math>\mathbb R \mbox{ is a V.S. over } \mathbb Q</math></li>		<li><math>\mathbb R \mbox{ is a V.S. over } \mathbb Q</math></li>
	<li><math>\mathbb R \mbox{ is a V.S. over } \mathbb R</math></li>		<li><math>\mathbb R \mbox{ is a V.S. over } \mathbb R</math></li>
	<li><math>\{0\} \mbox{ is a V.S. over } F</math></li>		<li><math>\,\! \{0\} \mbox{ is a V.S. over } F</math></li>
	</ol>		</ol>

C8sd: /* Examples */ Forgot $tags.$

2009-10-02T22:23:34Z

Examples: Forgot <math> tags.

← Older revision		Revision as of 18:23, 2 October 2009
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	'''Claim''': + is associative. Given <math>f, g, h \in \mathcal F(S, F), (f + g) + h = f + (g + h) \forall \sigma</math>		'''Claim''': + is associative. Given <math>f, g, h \in \mathcal F(S, F), (f + g) + h = f + (g + h) \forall \sigma</math>
	: ((f + g) + h)(\sigma) = (f + g)(\sigma) + h(\sigma)		: <math>((f + g) + h)(\sigma) = (f + g)(\sigma) + h(\sigma)</math>
	: = (f(\sigma) + g(\sigma)) + h(\sigma)		: <math>= (f(\sigma) + g(\sigma)) + h(\sigma)</math>
	: = f(\sigma) + g(\sigma) + h(\sigma) \mbox{ (by F2)}		: <math>= f(\sigma) + g(\sigma) + h(\sigma) \mbox{ (by F2)}</math>
	: (f + (g + h))(\sigma) = f(\sigma) + (g + h)(\sigma)		: <math>(f + (g + h))(\sigma) = f(\sigma) + (g + h)(\sigma)</math>
	: = f(\sigma) + (g(\sigma) + h(\sigma))		: <math>= f(\sigma) + (g(\sigma) + h(\sigma))</math>
	: = f(\sigma) + g(\sigma) + h(\sigma)		: <math>= f(\sigma) + g(\sigma) + h(\sigma)</math>

	<ol start="4">		<ol start="4">

C8sd at 22:18, 2 October 2009

2009-10-02T22:18:36Z

New page

{{09-240/Navigation}}

{{09-240/Class Notes Warning}}

{| style="border: solid 1px black"
|-
| Convention for today:
: <math>x, y, z \in \mathbf V</math>
: <math>a, b, c \in F</math>
|}

A V.S. over <math>F: V, 0, +, \times</math> s.t.

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

=== Examples ===

# <math>\left\{ \begin{pmatrix} a_1 \\ \vdots \\ a_n \end{pmatrix} : a_i \in F \right\} = F^n</math>
# <math>\left\{ \begin{pmatrix}
a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & a_{mn}
\end{pmatrix} \right\} = \mathrm M_{m \times n}(F)</math>
# Let ''S'' be a set (''F'' is some field)
#: <math>\mathcal F(S, F) = \{f: S \rightarrow F\}</math>
#: ''S'' = Primary colours = {red, green, blue}
#: ''F'' = ''F''2 = {0, 1}
#: <math>\mathcal F(S, F) = \left\{ \begin{matrix}
f_1(red) = 0 & f_1(green) = 1 & f_1(blue) = 0 \\
\cdots \\
f_2 \begin{pmatrix} \mbox{red} \\ \mbox{green} \\ \mbox{blue} \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} & \cdots
\end{matrix} \right\}</math>

----

<math>S = \mathbb N = \{ 1, 2, 3, 4, \ldots \} F = \mathbb R</math>

<math>\mathcal F(\mathbb N, \mathbb R) = \left\{ \begin{matrix}
1 & 2 & 3 & 4 & \ldots \\
6 & 6 & 6 & 6 & \ldots \\
\pi & 2\pi & e & 62 & \ldots \\
\end{matrix} \right\} = \{ \mbox{sequences} \}</math>

<math>S = \begin{pmatrix} \vdots \\ \vdots \end{pmatrix} n \Rightarrow \mathcal F(S, F) = F^n</math>

# <math>O_{\mathcal F(S, F)}(\sigma) = 0_F \forall \sigma \in S</math>
# <math>f, g \in \mathcal F(S, F)</math>
#: <math>(f + g)(\sigma) = f(\sigma) + g(\sigma)</math>
#: <math>f \in \mathcal F(S, F)</math>
#: <math>a \in F \Rightarrow \forall \sigma \in S, S(af)(\sigma) = a \cdot (f(\sigma))</math>

'''Claim''': + is associative. Given <math>f, g, h \in \mathcal F(S, F), (f + g) + h = f + (g + h) \forall \sigma</math>
: ((f + g) + h)(\sigma) = (f + g)(\sigma) + h(\sigma)
: = (f(\sigma) + g(\sigma)) + h(\sigma)
: = f(\sigma) + g(\sigma) + h(\sigma) \mbox{ (by F2)}
: (f + (g + h))(\sigma) = f(\sigma) + (g + h)(\sigma)
: = f(\sigma) + (g(\sigma) + h(\sigma))
: = f(\sigma) + g(\sigma) + h(\sigma)

<ol start="4">
<li><math>\mathbb C \mbox{ is a V.S. over } \mathbb R</math></li>
<li><math>\mathbb R \mbox{ is a V.S. over } \mathbb Q</math></li>
<li><math>\mathbb R \mbox{ is a V.S. over } \mathbb R</math></li>
<li><math>\{0\} \mbox{ is a V.S. over } F</math></li>
</ol>

=== Dull theorem ===

# Cancellation: <math>x + y = x + z \Rightarrow y = z</math> (add ''w'' to both sides s.t. ''x'' + ''w'' = 0)
# 0 is unique
# Negatives are unique: <math>x + y = 0 = x + z \Rightarrow y = 0</math>
# <math>0x = 0. a \cdot 0 = 0</math>
# <math>(-a) \cdot x = a \cdot (-x) = -(ax)</math>

09-240/Classnotes for Thursday September 24 - Revision history

Raymct at 02:47, 22 October 2009

Raymct at 02:44, 22 October 2009

C8sd: /* Examples */ Fix indentation, forced png.

C8sd: /* Examples */ Forgot tags.

C8sd at 22:18, 2 October 2009

C8sd: /* Examples */ Forgot $tags.$