06-240/Classnotes For Thursday, September 21 - Revision history

Drorbn: /* Force Vectors */

2007-07-11T11:42:41Z

Force Vectors

← Older revision		Revision as of 07:42, 11 July 2007
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	==Force Vectors==		==Force Vectors==
	A force has a direction and a magnitude.		A force has a direction and a magnitude.
	#~~<math>\mbox{~~There is a special force vector called 0.~~}</math>~~		# There is a special force vector called 0.
	#~~<math>\mbox{~~They can be added.~~}</math>~~		# They can be added.
	#~~<math>\mbox{~~They can be multiplied by any scalar.~~}</math>~~		# They can be multiplied by any scalar.

	====''Properties''====		====''Properties''====

	~~<math>\mbox{~~(convention: }x,y,z~~\mbox{ }\mbox{~~ are vectors; }a,b,c~~\mbox{ }\mbox{~~ are scalars)~~}</math>~~		(convention: <math>x,y,z</math> are vectors; <math>a,b,c</math> are scalars)
	#<math> x+y=y+x \ </math>		#<math> x+y=y+x </math>
	#<math> x+(y+z)=(x+y)+z \ </math>		#<math> x+(y+z)=(x+y)+z \ </math>
	#<math> x+0=x \ </math>		#<math> x+0=x \ </math>

Drorbn: Reverted edit of 74.97.115.74, changed back to last version by Drorbn

2007-06-14T15:11:15Z

Reverted edit of 74.97.115.74, changed back to last version by Drorbn

← Older revision		Revision as of 11:11, 14 June 2007
Line 20:		Line 20:

	<math>\mbox{(convention: }x,y,z\mbox{ }\mbox{ are vectors; }a,b,c\mbox{ }\mbox{ are scalars)}</math>		<math>\mbox{(convention: }x,y,z\mbox{ }\mbox{ are vectors; }a,b,c\mbox{ }\mbox{ are scalars)}</math>
	#<math> x y=y x \ </math>		#<math> x+y=y+x \ </math>
	#<math> x (y z)=(x y) z \ </math>		#<math> x+(y+z)=(x+y)+z \ </math>
	#<math> x 0=x \ </math>		#<math> x+0=x \ </math>
	#<math> \forall x\; \exists\ y \ \mbox{ s.t. }x y=0</math>		#<math> \forall x\; \exists\ y \ \mbox{ s.t. }x+y=0</math>
	#<math> 1\cdot x=x \ </math>		#<math> 1\cdot x=x \ </math>
	#<math> a(bx)=(ab)x \ </math>		#<math> a(bx)=(ab)x \ </math>
	#<math> a(x y)=ax ay \ </math>		#<math> a(x+y)=ax+ay \ </math>
	#<math> (a b)x=ax bx \ </math>		#<math> (a+b)x=ax+bx \ </math>

	=====Definition=====		=====Definition=====
Line 33:		Line 33:
	Let F be a field "of scalars". A vector space over F is a set V, of "vectors", along with two operations		Let F be a field "of scalars". A vector space over F is a set V, of "vectors", along with two operations

	: <math> : V \times V \to V </math>		: <math> +: V \times V \to V </math>
	: <math> \cdot: F \times V \to V \mbox{, so that:}</math>		: <math> \cdot: F \times V \to V \mbox{, so that:}</math>
	#<math> \forall x,y \in V\ x y=y x </math>		#<math> \forall x,y \in V\ x+y=y+x </math>
	#<math> \forall x,y \in V\ x (y z)=(x y) z </math>		#<math> \forall x,y \in V\ x+(y+z)=(x+y)+z </math>
	#<math> \exists\ 0 \in V s.t.\; \forall x \in V\ x 0=x </math>		#<math> \exists\ 0 \in V s.t.\; \forall x \in V\ x+0=x </math>
	#<math> \forall x \in V\; \exists\ y \in V\ s.t. \ x y=0</math>		#<math> \forall x \in V\; \exists\ y \in V\ s.t. \ x+y=0</math>
	#<math> 1\cdot x=x\ </math>		#<math> 1\cdot x=x\ </math>
	#<math> a(bx)=(ab)x\ </math>		#<math> a(bx)=(ab)x\ </math>
	#<math> a(x y)=ax ay\ </math>		#<math> a(x+y)=ax+ay\ </math>
	#<math> \forall x \in V\ ,\forall a,b \in F\ (a b)x=ax bx </math>		#<math> \forall x \in V\ ,\forall a,b \in F\ (a+b)x=ax+bx </math>
	-----		-----
	9. <math> x \mapsto \vert x\vert \in \mathbb{R} \ \vert x y\vert \le \vert x\vert \vert y\vert </math>		9. <math> x \mapsto \vert x\vert \in \mathbb{R} \ \vert x+y\vert \le \vert x\vert+\vert y\vert </math>
	====''Examples''====		====''Examples''====
	'''Ex.1.'''		'''Ex.1.'''
Line 50:		Line 50:
	<math> n \in \mathbb{Z}\ , n \ge 0 </math> <br/>		<math> n \in \mathbb{Z}\ , n \ge 0 </math> <br/>
	<math> x=(a_1,\ldots,a_2)\ y=(b_1,\ldots, b_2)\ </math> <br/>		<math> x=(a_1,\ldots,a_2)\ y=(b_1,\ldots, b_2)\ </math> <br/>
	<math> x y:=(a_1 b_1,a_2 b_2,\ldots,a_n b_n)\ </math> <br/>		<math> x+y:=(a_1+b_1,a_2+b_2,\ldots,a_n+b_n)\ </math> <br/>
	<math> 0_{F^n}=(0,\ldots,0) </math> <br/>		<math> 0_{F^n}=(0,\ldots,0) </math> <br/>
	<math> a\in F\ ax=(aa_1,aa_2,\ldots,aa_n) </math> <br/>		<math> a\in F\ ax=(aa_1,aa_2,\ldots,aa_n) </math> <br/>
	<math> \mbox{In } \mathbb{Q}^3 \ \left( \frac{3}{2},-2,7\right) \left( \frac{-3}{2}, \frac{1}{3},240\right)=\left(0, \frac{-5}{3},247\right) </math> <br/>		<math> \mbox{In } \mathbb{Q}^3 \ \left( \frac{3}{2},-2,7\right)+\left( \frac{-3}{2}, \frac{1}{3},240\right)=\left(0, \frac{-5}{3},247\right) </math> <br/>
	<math> 7\left( \frac{1}{5},\frac{1}{7},\frac{1}{9}\right)=\left( \frac{7}{5},1,\frac{7}{9}\right) </math> <br/>		<math> 7\left( \frac{1}{5},\frac{1}{7},\frac{1}{9}\right)=\left( \frac{7}{5},1,\frac{7}{9}\right) </math> <br/>
	'''Ex.2.'''		'''Ex.2.'''
	<math> V=M_{m\times n}(F)=\left\lbrace\begin{pmatrix} a_{11}~~</math>~~		<math> V=M_{m\times n}(F)=\left\lbrace\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots &
			& \vdots \\ a_{m1} & \cdots & a_{mn}\end{pmatrix}: a_{ij} \in F \right\rbrace </math> <br/>
			<math> M_{3\times 2}( \mathbb{R})\ni \begin{pmatrix} 7 & -7 \\ \pi & \mathit{e} \\ -5 & 2 \end{pmatrix} </math> <br/>
			<math>\mbox{Addition by adding entry by entry:}</math>

			<math> M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}+\begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}=\begin{pmatrix} {a_{11}+b_{11}} & {a_{12}+b_{12}} \\ {a_{21}+b_{21}} & {a_{22}+b_{22}} \end{pmatrix}</math> <br/>

			<math>\mbox{Multiplication by multiplying scalar c to all entries by M.}</math>

			<math> c\cdot M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}=\begin{pmatrix} c\cdot a_{11} & c\cdot a_{12} \\ c\cdot a_{21} & c\cdot a_{22} \end{pmatrix}</math> <br/> <br/>

			<math>\mbox{Zero matrix has all entries = 0:}</math>

			<math> 0_{M_{m\times n}}=\begin{pmatrix} 0 & \cdots & 0 \\ \vdots &
			& \vdots \\ 0 & \cdots & 0\end{pmatrix} </math> <br/>
			'''Ex.3.'''
			<math> \mathbb{C}</math> form a vector space over <math> \mathbb{R}</math>. <br/>
			'''Ex.4.'''
			<math>\mbox{F is a vector space over itself.}</math> <br/>
			'''Ex.5.'''
			<math> \mathbb{R}</math> is a vector space over <math> \mathbb{Q}</math>. <br/>
			'''Ex.6.'''
			<math>\mbox{Let S be a set. Let}</math> <br/>
			<math> \mathcal{F}(S,\mathbb{R})=\big\{f:S\to \mathbb{R} \big\} </math> <br/>
			<math> f,g \in \mathcal{F}(S,\mathbb{R}) </math> <br/>
			<math> (f+g)(t)=f(t)+g(t)\ for\ any\ t\in S </math> <br/>
			<math> (af)(t)=a\cdot f(t)\ </math>

74.97.115.74 at 18:52, 13 June 2007

2007-06-13T18:52:23Z

← Older revision		Revision as of 14:52, 13 June 2007
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	<math>\mbox{(convention: }x,y,z\mbox{ }\mbox{ are vectors; }a,b,c\mbox{ }\mbox{ are scalars)}</math>		<math>\mbox{(convention: }x,y,z\mbox{ }\mbox{ are vectors; }a,b,c\mbox{ }\mbox{ are scalars)}</math>
	#<math> x+y=y+x \ </math>		#<math> x y=y x \ </math>
	#<math> x+(y+z)=(x+y)+z \ </math>		#<math> x (y z)=(x y) z \ </math>
	#<math> x+0=x \ </math>		#<math> x 0=x \ </math>
	#<math> \forall x\; \exists\ y \ \mbox{ s.t. }x+y=0</math>		#<math> \forall x\; \exists\ y \ \mbox{ s.t. }x y=0</math>
	#<math> 1\cdot x=x \ </math>		#<math> 1\cdot x=x \ </math>
	#<math> a(bx)=(ab)x \ </math>		#<math> a(bx)=(ab)x \ </math>
	#<math> a(x+y)=ax+ay \ </math>		#<math> a(x y)=ax ay \ </math>
	#<math> (a+b)x=ax+bx \ </math>		#<math> (a b)x=ax bx \ </math>

	=====Definition=====		=====Definition=====
Line 33:		Line 33:
	Let F be a field "of scalars". A vector space over F is a set V, of "vectors", along with two operations		Let F be a field "of scalars". A vector space over F is a set V, of "vectors", along with two operations

	: <math> +: V \times V \to V </math>		: <math> : V \times V \to V </math>
	: <math> \cdot: F \times V \to V \mbox{, so that:}</math>		: <math> \cdot: F \times V \to V \mbox{, so that:}</math>
	#<math> \forall x,y \in V\ x+y=y+x </math>		#<math> \forall x,y \in V\ x y=y x </math>
	#<math> \forall x,y \in V\ x+(y+z)=(x+y)+z </math>		#<math> \forall x,y \in V\ x (y z)=(x y) z </math>
	#<math> \exists\ 0 \in V s.t.\; \forall x \in V\ x+0=x </math>		#<math> \exists\ 0 \in V s.t.\; \forall x \in V\ x 0=x </math>
	#<math> \forall x \in V\; \exists\ y \in V\ s.t. \ x+y=0</math>		#<math> \forall x \in V\; \exists\ y \in V\ s.t. \ x y=0</math>
	#<math> 1\cdot x=x\ </math>		#<math> 1\cdot x=x\ </math>
	#<math> a(bx)=(ab)x\ </math>		#<math> a(bx)=(ab)x\ </math>
	#<math> a(x+y)=ax+ay\ </math>		#<math> a(x y)=ax ay\ </math>
	#<math> \forall x \in V\ ,\forall a,b \in F\ (a+b)x=ax+bx </math>		#<math> \forall x \in V\ ,\forall a,b \in F\ (a b)x=ax bx </math>
	-----		-----
	9. <math> x \mapsto \vert x\vert \in \mathbb{R} \ \vert x+y\vert \le \vert x\vert+\vert y\vert </math>		9. <math> x \mapsto \vert x\vert \in \mathbb{R} \ \vert x y\vert \le \vert x\vert \vert y\vert </math>
	====''Examples''====		====''Examples''====
	'''Ex.1.'''		'''Ex.1.'''
Line 50:		Line 50:
	<math> n \in \mathbb{Z}\ , n \ge 0 </math> <br/>		<math> n \in \mathbb{Z}\ , n \ge 0 </math> <br/>
	<math> x=(a_1,\ldots,a_2)\ y=(b_1,\ldots, b_2)\ </math> <br/>		<math> x=(a_1,\ldots,a_2)\ y=(b_1,\ldots, b_2)\ </math> <br/>
	<math> x+y:=(a_1+b_1,a_2+b_2,\ldots,a_n+b_n)\ </math> <br/>		<math> x y:=(a_1 b_1,a_2 b_2,\ldots,a_n b_n)\ </math> <br/>
	<math> 0_{F^n}=(0,\ldots,0) </math> <br/>		<math> 0_{F^n}=(0,\ldots,0) </math> <br/>
	<math> a\in F\ ax=(aa_1,aa_2,\ldots,aa_n) </math> <br/>		<math> a\in F\ ax=(aa_1,aa_2,\ldots,aa_n) </math> <br/>
	<math> \mbox{In } \mathbb{Q}^3 \ \left( \frac{3}{2},-2,7\right)+\left( \frac{-3}{2}, \frac{1}{3},240\right)=\left(0, \frac{-5}{3},247\right) </math> <br/>		<math> \mbox{In } \mathbb{Q}^3 \ \left( \frac{3}{2},-2,7\right) \left( \frac{-3}{2}, \frac{1}{3},240\right)=\left(0, \frac{-5}{3},247\right) </math> <br/>
	<math> 7\left( \frac{1}{5},\frac{1}{7},\frac{1}{9}\right)=\left( \frac{7}{5},1,\frac{7}{9}\right) </math> <br/>		<math> 7\left( \frac{1}{5},\frac{1}{7},\frac{1}{9}\right)=\left( \frac{7}{5},1,\frac{7}{9}\right) </math> <br/>
	'''Ex.2.'''		'''Ex.2.'''
	<math> V=M_{m\times n}(F)=\left\lbrace\begin{pmatrix} a_{11} ~~& \cdots & a_{1n} \\ \vdots &~~		<math> V=M_{m\times n}(F)=\left\lbrace\begin{pmatrix} a_{11}</math>
	& \vdots \\ a_{m1} & \cdots & a_{mn}\end{pmatrix}: a_{ij} \in F \right\rbrace </math> <br/>
	<math> M_{3\times 2}( \mathbb{R})\ni \begin{pmatrix} 7 & -7 \\ \pi & \mathit{e} \\ -5 & 2 \end{pmatrix} </math> <br/>
	<math>\mbox{Addition by adding entry by entry:}</math>

	<math> M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}+\begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}=\begin{pmatrix} {a_{11}+b_{11}} & {a_{12}+b_{12}} \\ {a_{21}+b_{21}} & {a_{22}+b_{22}} \end{pmatrix}</math> <br/>

	<math>\mbox{Multiplication by multiplying scalar c to all entries by M.}</math>

	<math> c\cdot M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}=\begin{pmatrix} c\cdot a_{11} & c\cdot a_{12} \\ c\cdot a_{21} & c\cdot a_{22} \end{pmatrix}</math> <br/> <br/>

	<math>\mbox{Zero matrix has all entries = 0:}</math>

	<math> 0_{M_{m\times n}}=\begin{pmatrix} 0 & \cdots & 0 \\ \vdots &
	& \vdots \\ 0 & \cdots & 0\end{pmatrix} </math> <br/>
	'''Ex.3.'''
	<math> \mathbb{C}</math> form a vector space over <math> \mathbb{R}</math>. <br/>
	'''Ex.4.'''
	<math>\mbox{F is a vector space over itself.}</math> <br/>
	'''Ex.5.'''
	<math> \mathbb{R}</math> is a vector space over <math> \mathbb{Q}</math>. <br/>
	'''Ex.6.'''
	<math>\mbox{Let S be a set. Let}</math> <br/>
	<math> \mathcal{F}(S,\mathbb{R})=\big\{f:S\to \mathbb{R} \big\} </math> <br/>
	<math> f,g \in \mathcal{F}(S,\mathbb{R}) </math> <br/>
	<math> (f+g)(t)=f(t)+g(t)\ for\ any\ t\in S </math> <br/>
	<math> (af)(t)=a\cdot f(t)\ </math>

Drorbn: Reverted edit of 193.232.119.186, changed back to last version by Wongpak

2007-05-28T14:51:02Z

Reverted edit of 193.232.119.186, changed back to last version by Wongpak

← Older revision		Revision as of 10:51, 28 May 2007
Line 20:		Line 20:

	<math>\mbox{(convention: }x,y,z\mbox{ }\mbox{ are vectors; }a,b,c\mbox{ }\mbox{ are scalars)}</math>		<math>\mbox{(convention: }x,y,z\mbox{ }\mbox{ are vectors; }a,b,c\mbox{ }\mbox{ are scalars)}</math>
	#<math> x y=y x \ </math>		#<math> x+y=y+x \ </math>
	#<math> x (y z)=(x y) z \ </math>		#<math> x+(y+z)=(x+y)+z \ </math>
	#<math> x 0=x \ </math>		#<math> x+0=x \ </math>
	#<math> \forall x\; \exists\ y \ \mbox{ s.t. }x y=0</math>		#<math> \forall x\; \exists\ y \ \mbox{ s.t. }x+y=0</math>
	#<math> 1\cdot x=x \ </math>		#<math> 1\cdot x=x \ </math>
	#<math> a(bx)=(ab)x \ </math>		#<math> a(bx)=(ab)x \ </math>
	#<math> a(x y)=ax ay \ </math>		#<math> a(x+y)=ax+ay \ </math>
	#<math> (a b)x=ax bx \ </math>		#<math> (a+b)x=ax+bx \ </math>

	=====Definition=====		=====Definition=====
Line 33:		Line 33:
	Let F be a field "of scalars". A vector space over F is a set V, of "vectors", along with two operations		Let F be a field "of scalars". A vector space over F is a set V, of "vectors", along with two operations

	: <math> : V \times V \to V </math>		: <math> +: V \times V \to V </math>
	: <math> \cdot: F \times V \to V \mbox{, so that:}</math>		: <math> \cdot: F \times V \to V \mbox{, so that:}</math>
	#<math> \forall x,y \in V\ x y=y x </math>		#<math> \forall x,y \in V\ x+y=y+x </math>
	#<math> \forall x,y \in V\ x (y z)=(x y) z </math>		#<math> \forall x,y \in V\ x+(y+z)=(x+y)+z </math>
	#<math> \exists\ 0 \in V s.t.\; \forall x \in V\ x 0=x </math>		#<math> \exists\ 0 \in V s.t.\; \forall x \in V\ x+0=x </math>
	#<math> \forall x \in V\; \exists\ y \in V\ s.t. \ x y=0</math>		#<math> \forall x \in V\; \exists\ y \in V\ s.t. \ x+y=0</math>
	#<math> 1\cdot x=x\ </math>		#<math> 1\cdot x=x\ </math>
	#<math> a(bx)=(ab)x\ </math>		#<math> a(bx)=(ab)x\ </math>
	#<math> a(x y)=ax ay\ </math>		#<math> a(x+y)=ax+ay\ </math>
	#<math> \forall x \in V\ ,\forall a,b \in F\ (a b)x=ax bx </math>		#<math> \forall x \in V\ ,\forall a,b \in F\ (a+b)x=ax+bx </math>
	-----		-----
	9. <math> x \mapsto \vert x\vert \in \mathbb{R} \ \vert x y\vert \le \vert x\vert \vert y\vert </math>		9. <math> x \mapsto \vert x\vert \in \mathbb{R} \ \vert x+y\vert \le \vert x\vert+\vert y\vert </math>
	====''Examples''====		====''Examples''====
	'''Ex.1.'''		'''Ex.1.'''
Line 50:		Line 50:
	<math> n \in \mathbb{Z}\ , n \ge 0 </math> <br/>		<math> n \in \mathbb{Z}\ , n \ge 0 </math> <br/>
	<math> x=(a_1,\ldots,a_2)\ y=(b_1,\ldots, b_2)\ </math> <br/>		<math> x=(a_1,\ldots,a_2)\ y=(b_1,\ldots, b_2)\ </math> <br/>
	<math> x y:=(a_1 b_1,a_2 b_2,\ldots,a_n b_n)\ </math> <br/>		<math> x+y:=(a_1+b_1,a_2+b_2,\ldots,a_n+b_n)\ </math> <br/>
	<math> 0_{F^n}=(0,\ldots,0) </math> <br/>		<math> 0_{F^n}=(0,\ldots,0) </math> <br/>
	<math> a\in F\ ax=(aa_1,aa_2,\ldots,aa_n) </math> <br/>		<math> a\in F\ ax=(aa_1,aa_2,\ldots,aa_n) </math> <br/>
	<math> \mbox{In } \mathbb{Q}^3 \ \left( \frac{3}{2},-2,7\right) \left( \frac{-3}{2}, \frac{1}{3},240\right)=\left(0, \frac{-5}{3},247\right) </math> <br/>		<math> \mbox{In } \mathbb{Q}^3 \ \left( \frac{3}{2},-2,7\right)+\left( \frac{-3}{2}, \frac{1}{3},240\right)=\left(0, \frac{-5}{3},247\right) </math> <br/>
	<math> 7\left( \frac{1}{5},\frac{1}{7},\frac{1}{9}\right)=\left( \frac{7}{5},1,\frac{7}{9}\right) </math> <br/>		<math> 7\left( \frac{1}{5},\frac{1}{7},\frac{1}{9}\right)=\left( \frac{7}{5},1,\frac{7}{9}\right) </math> <br/>
	'''Ex.2.'''		'''Ex.2.'''
	<math> V=M_{m\times n}(F)=\left\lbrace\begin{pmatrix} a_{11}~~</math>~~		<math> V=M_{m\times n}(F)=\left\lbrace\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots &
			& \vdots \\ a_{m1} & \cdots & a_{mn}\end{pmatrix}: a_{ij} \in F \right\rbrace </math> <br/>
			<math> M_{3\times 2}( \mathbb{R})\ni \begin{pmatrix} 7 & -7 \\ \pi & \mathit{e} \\ -5 & 2 \end{pmatrix} </math> <br/>
			<math>\mbox{Addition by adding entry by entry:}</math>

			<math> M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}+\begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}=\begin{pmatrix} {a_{11}+b_{11}} & {a_{12}+b_{12}} \\ {a_{21}+b_{21}} & {a_{22}+b_{22}} \end{pmatrix}</math> <br/>

			<math>\mbox{Multiplication by multiplying scalar c to all entries by M.}</math>

			<math> c\cdot M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}=\begin{pmatrix} c\cdot a_{11} & c\cdot a_{12} \\ c\cdot a_{21} & c\cdot a_{22} \end{pmatrix}</math> <br/> <br/>

			<math>\mbox{Zero matrix has all entries = 0:}</math>

			<math> 0_{M_{m\times n}}=\begin{pmatrix} 0 & \cdots & 0 \\ \vdots &
			& \vdots \\ 0 & \cdots & 0\end{pmatrix} </math> <br/>
			'''Ex.3.'''
			<math> \mathbb{C}</math> form a vector space over <math> \mathbb{R}</math>. <br/>
			'''Ex.4.'''
			<math>\mbox{F is a vector space over itself.}</math> <br/>
			'''Ex.5.'''
			<math> \mathbb{R}</math> is a vector space over <math> \mathbb{Q}</math>. <br/>
			'''Ex.6.'''
			<math>\mbox{Let S be a set. Let}</math> <br/>
			<math> \mathcal{F}(S,\mathbb{R})=\big\{f:S\to \mathbb{R} \big\} </math> <br/>
			<math> f,g \in \mathcal{F}(S,\mathbb{R}) </math> <br/>
			<math> (f+g)(t)=f(t)+g(t)\ for\ any\ t\in S </math> <br/>
			<math> (af)(t)=a\cdot f(t)\ </math>

193.232.119.186 at 06:40, 28 May 2007

2007-05-28T06:40:43Z

← Older revision		Revision as of 02:40, 28 May 2007
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	<math>\mbox{(convention: }x,y,z\mbox{ }\mbox{ are vectors; }a,b,c\mbox{ }\mbox{ are scalars)}</math>		<math>\mbox{(convention: }x,y,z\mbox{ }\mbox{ are vectors; }a,b,c\mbox{ }\mbox{ are scalars)}</math>
	#<math> x+y=y+x \ </math>		#<math> x y=y x \ </math>
	#<math> x+(y+z)=(x+y)+z \ </math>		#<math> x (y z)=(x y) z \ </math>
	#<math> x+0=x \ </math>		#<math> x 0=x \ </math>
	#<math> \forall x\; \exists\ y \ \mbox{ s.t. }x+y=0</math>		#<math> \forall x\; \exists\ y \ \mbox{ s.t. }x y=0</math>
	#<math> 1\cdot x=x \ </math>		#<math> 1\cdot x=x \ </math>
	#<math> a(bx)=(ab)x \ </math>		#<math> a(bx)=(ab)x \ </math>
	#<math> a(x+y)=ax+ay \ </math>		#<math> a(x y)=ax ay \ </math>
	#<math> (a+b)x=ax+bx \ </math>		#<math> (a b)x=ax bx \ </math>

	=====Definition=====		=====Definition=====
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	Let F be a field "of scalars". A vector space over F is a set V, of "vectors", along with two operations		Let F be a field "of scalars". A vector space over F is a set V, of "vectors", along with two operations

	: <math> +: V \times V \to V </math>		: <math> : V \times V \to V </math>
	: <math> \cdot: F \times V \to V \mbox{, so that:}</math>		: <math> \cdot: F \times V \to V \mbox{, so that:}</math>
	#<math> \forall x,y \in V\ x+y=y+x </math>		#<math> \forall x,y \in V\ x y=y x </math>
	#<math> \forall x,y \in V\ x+(y+z)=(x+y)+z </math>		#<math> \forall x,y \in V\ x (y z)=(x y) z </math>
	#<math> \exists\ 0 \in V s.t.\; \forall x \in V\ x+0=x </math>		#<math> \exists\ 0 \in V s.t.\; \forall x \in V\ x 0=x </math>
	#<math> \forall x \in V\; \exists\ y \in V\ s.t. \ x+y=0</math>		#<math> \forall x \in V\; \exists\ y \in V\ s.t. \ x y=0</math>
	#<math> 1\cdot x=x\ </math>		#<math> 1\cdot x=x\ </math>
	#<math> a(bx)=(ab)x\ </math>		#<math> a(bx)=(ab)x\ </math>
	#<math> a(x+y)=ax+ay\ </math>		#<math> a(x y)=ax ay\ </math>
	#<math> \forall x \in V\ ,\forall a,b \in F\ (a+b)x=ax+bx </math>		#<math> \forall x \in V\ ,\forall a,b \in F\ (a b)x=ax bx </math>
	-----		-----
	9. <math> x \mapsto \vert x\vert \in \mathbb{R} \ \vert x+y\vert \le \vert x\vert+\vert y\vert </math>		9. <math> x \mapsto \vert x\vert \in \mathbb{R} \ \vert x y\vert \le \vert x\vert \vert y\vert </math>
	====''Examples''====		====''Examples''====
	'''Ex.1.'''		'''Ex.1.'''
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	<math> n \in \mathbb{Z}\ , n \ge 0 </math> <br/>		<math> n \in \mathbb{Z}\ , n \ge 0 </math> <br/>
	<math> x=(a_1,\ldots,a_2)\ y=(b_1,\ldots, b_2)\ </math> <br/>		<math> x=(a_1,\ldots,a_2)\ y=(b_1,\ldots, b_2)\ </math> <br/>
	<math> x+y:=(a_1+b_1,a_2+b_2,\ldots,a_n+b_n)\ </math> <br/>		<math> x y:=(a_1 b_1,a_2 b_2,\ldots,a_n b_n)\ </math> <br/>
	<math> 0_{F^n}=(0,\ldots,0) </math> <br/>		<math> 0_{F^n}=(0,\ldots,0) </math> <br/>
	<math> a\in F\ ax=(aa_1,aa_2,\ldots,aa_n) </math> <br/>		<math> a\in F\ ax=(aa_1,aa_2,\ldots,aa_n) </math> <br/>
	<math> \mbox{In } \mathbb{Q}^3 \ \left( \frac{3}{2},-2,7\right)+\left( \frac{-3}{2}, \frac{1}{3},240\right)=\left(0, \frac{-5}{3},247\right) </math> <br/>		<math> \mbox{In } \mathbb{Q}^3 \ \left( \frac{3}{2},-2,7\right) \left( \frac{-3}{2}, \frac{1}{3},240\right)=\left(0, \frac{-5}{3},247\right) </math> <br/>
	<math> 7\left( \frac{1}{5},\frac{1}{7},\frac{1}{9}\right)=\left( \frac{7}{5},1,\frac{7}{9}\right) </math> <br/>		<math> 7\left( \frac{1}{5},\frac{1}{7},\frac{1}{9}\right)=\left( \frac{7}{5},1,\frac{7}{9}\right) </math> <br/>
	'''Ex.2.'''		'''Ex.2.'''
	<math> V=M_{m\times n}(F)=\left\lbrace\begin{pmatrix} a_{11} ~~& \cdots & a_{1n} \\ \vdots &~~		<math> V=M_{m\times n}(F)=\left\lbrace\begin{pmatrix} a_{11}</math>
	& \vdots \\ a_{m1} & \cdots & a_{mn}\end{pmatrix}: a_{ij} \in F \right\rbrace </math> <br/>
	<math> M_{3\times 2}( \mathbb{R})\ni \begin{pmatrix} 7 & -7 \\ \pi & \mathit{e} \\ -5 & 2 \end{pmatrix} </math> <br/>
	<math>\mbox{Addition by adding entry by entry:}</math>

	<math> M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}+\begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}=\begin{pmatrix} {a_{11}+b_{11}} & {a_{12}+b_{12}} \\ {a_{21}+b_{21}} & {a_{22}+b_{22}} \end{pmatrix}</math> <br/>

	<math>\mbox{Multiplication by multiplying scalar c to all entries by M.}</math>

	<math> c\cdot M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}=\begin{pmatrix} c\cdot a_{11} & c\cdot a_{12} \\ c\cdot a_{21} & c\cdot a_{22} \end{pmatrix}</math> <br/> <br/>

	<math>\mbox{Zero matrix has all entries = 0:}</math>

	<math> 0_{M_{m\times n}}=\begin{pmatrix} 0 & \cdots & 0 \\ \vdots &
	& \vdots \\ 0 & \cdots & 0\end{pmatrix} </math> <br/>
	'''Ex.3.'''
	<math> \mathbb{C}</math> form a vector space over <math> \mathbb{R}</math>. <br/>
	'''Ex.4.'''
	<math>\mbox{F is a vector space over itself.}</math> <br/>
	'''Ex.5.'''
	<math> \mathbb{R}</math> is a vector space over <math> \mathbb{Q}</math>. <br/>
	'''Ex.6.'''
	<math>\mbox{Let S be a set. Let}</math> <br/>
	<math> \mathcal{F}(S,\mathbb{R})=\big\{f:S\to \mathbb{R} \big\} </math> <br/>
	<math> f,g \in \mathcal{F}(S,\mathbb{R}) </math> <br/>
	<math> (f+g)(t)=f(t)+g(t)\ for\ any\ t\in S </math> <br/>
	<math> (af)(t)=a\cdot f(t)\ </math>

Wongpak: /* Scan of Lecture Notes */

2006-10-25T02:34:18Z

Scan of Lecture Notes

← Older revision		Revision as of 22:34, 24 October 2006
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	* PDF file by [[User:Alla]]: [[Media:MAT_Lect004.pdf\|Week 2 Lecture 2 notes]]		* PDF file by [[User:Alla]]: [[Media:MAT_Lect004.pdf\|Week 2 Lecture 2 notes]]
	* PDF file by [[User:Gokmen]]: [[Media:06-240-Lecture-21-september.pdf\|Week 2 Lecture notes]]		* PDF file by [[User:Gokmen]]: [[Media:06-240-Lecture-21-september.pdf\|Week 2 Lecture 2 notes]]

	==Scan of Tutorial notes==		==Scan of Tutorial notes==

Wongpak: /* Scan of Lecture Notes */

2006-10-25T02:32:16Z

Scan of Lecture Notes

← Older revision		Revision as of 22:32, 24 October 2006
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	* PDF file by [[User:Alla]]: [[Media:MAT_Lect004.pdf\|Week 2 Lecture 2 notes]]		* PDF file by [[User:Alla]]: [[Media:MAT_Lect004.pdf\|Week 2 Lecture 2 notes]]
	* PDF file by [[User:Gokmen]]: [[Media:06-240-Lecture-21-september.pdf\|Week 2 ~~Tutorial~~ notes]]		* PDF file by [[User:Gokmen]]: [[Media:06-240-Lecture-21-september.pdf\|Week 2 Lecture notes]]

	==Scan of Tutorial notes==		==Scan of Tutorial notes==

Wongpak at 02:30, 25 October 2006

2006-10-25T02:30:41Z

@@ Line 1: / Line 1: @@
 {{06-240/Navigation}}
 #<math>\mbox{There is a special force vector called 0.}</math>
 #<math>\mbox{They can be added.}</math>
@@ Line 74: / Line 83: @@
 <math> (f+g)(t)=f(t)+g(t)\ for\ any\ t\in S </math> <br/>
 <math> (af)(t)=a\cdot f(t)\ </math>

Alla: Added scan of Lecture and Tutorial notes

2006-10-01T03:05:02Z

Added scan of Lecture and Tutorial notes

← Older revision		Revision as of 23:05, 30 September 2006
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	<math> (f+g)(t)=f(t)+g(t)\ for\ any\ t\in S </math> <br/>		<math> (f+g)(t)=f(t)+g(t)\ for\ any\ t\in S </math> <br/>
	<math> (af)(t)=a\cdot f(t)\ </math>		<math> (af)(t)=a\cdot f(t)\ </math>

			====''Scan of Lecture Notes''====

			* PDF file by [[User:Alla]]: [[Media:MAT_Lect004.pdf\|Week 2 Lecture 2 notes]]

			==<center><u>'''Scan of Tutorial notes'''</u></center>==

			* PDF file by [[User:Alla]]: [[Media:MAT_Tut002.pdf\|Week 2 Tutorial notes]]

Drorbn at 02:27, 29 September 2006

2006-09-29T02:27:13Z

← Older revision		Revision as of 22:27, 28 September 2006
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			{{06-240/Navigation}}

	A force has a direction and a magnitude.		A force has a direction and a magnitude.