06-240/Classnotes For Thursday, September 21: Difference between revisions

Latest revision as of 06:42, 11 July 2007

#	Week of...	Notes and Links
1	Sep 11	About, Tue, HW1, Putnam, Thu
2	Sep 18	Tue, HW2, Thu
3	Sep 25	Tue, HW3, Photo, Thu
4	Oct 2	Tue, HW4, Thu
5	Oct 9	Tue, HW5, Thu
6	Oct 16	Why?, Iso, Tue, Thu
7	Oct 23	Term Test, Thu (double)
8	Oct 30	Tue, HW6, Thu
9	Nov 6	Tue, HW7, Thu
10	Nov 13	Tue, HW8, Thu
11	Nov 20	Tue, HW9, Thu
12	Nov 27	Tue, HW10, Thu
13	Dec 4	On the final, Tue, Thu
F	Dec 11	Final: Dec 13 2-5PM at BN3, Exam Forum
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Scan of Lecture Notes

PDF file by User:Alla: Week 2 Lecture 2 notes
PDF file by User:Gokmen: Week 2 Lecture 2 notes

Scan of Tutorial notes

PDF file by User:Alla: Week 2 Tutorial notes
PDF file by User:Gokmen: Week 2 Tutorial notes

Force Vectors

A force has a direction and a magnitude.

There is a special force vector called 0.
They can be added.
They can be multiplied by any scalar.

Properties

(convention: $x,y,z$ are vectors; $a,b,c$ are scalars)

$x+y=y+x$
$x+(y+z)=(x+y)+z\$
$x+0=x\$
$\forall x\;\exists \ y\ {\mbox{ s.t. }}x+y=0$
$1\cdot x=x\$
$a(bx)=(ab)x\$
$a(x+y)=ax+ay\$
$(a+b)x=ax+bx\$

Definition

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

+:V\times V\to V

\cdot :F\times V\to V{\mbox{, so that:}}

$\forall x,y\in V\ x+y=y+x$
$\forall x,y\in V\ x+(y+z)=(x+y)+z$
$\exists \ 0\in Vs.t.\;\forall x\in V\ x+0=x$
$\forall x\in V\;\exists \ y\in V\ s.t.\ x+y=0$
$1\cdot x=x\$
$a(bx)=(ab)x\$
$a(x+y)=ax+ay\$
$\forall x\in V\ ,\forall a,b\in F\ (a+b)x=ax+bx$

9. $x\mapsto \vert x\vert \in \mathbb {R} \ \vert x+y\vert \leq \vert x\vert +\vert y\vert$

Examples

Ex.1. $F^{n}=\lbrace (a_{1},a_{2},a_{3},\ldots ,a_{n-1},a_{n}):\forall i\ a_{i}\in F\rbrace$
$n\in \mathbb {Z} \ ,n\geq 0$
$x=(a_{1},\ldots ,a_{2})\ y=(b_{1},\ldots ,b_{2})\$
$x+y:=(a_{1}+b_{1},a_{2}+b_{2},\ldots ,a_{n}+b_{n})\$
$0_{F^{n}}=(0,\ldots ,0)$
$a\in F\ ax=(aa_{1},aa_{2},\ldots ,aa_{n})$
${\mbox{In }}\mathbb {Q} ^{3}\ \left({\frac {3}{2}},-2,7\right)+\left({\frac {-3}{2}},{\frac {1}{3}},240\right)=\left(0,{\frac {-5}{3}},247\right)$
$7\left({\frac {1}{5}},{\frac {1}{7}},{\frac {1}{9}}\right)=\left({\frac {7}{5}},1,{\frac {7}{9}}\right)$
Ex.2. $V=M_{m\times n}(F)=\left\lbrace {\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &&\vdots \\a_{m1}&\cdots &a_{mn}\end{pmatrix}}:a_{ij}\in F\right\rbrace$
$M_{3\times 2}(\mathbb {R} )\ni {\begin{pmatrix}7&-7\\\pi &{\mathit {e}}\\-5&2\end{pmatrix}}$
${\mbox{Addition by adding entry by entry:}}$

$M_{2\times 2}\ {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}+{\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}={\begin{pmatrix}{a_{11}+b_{11}}&{a_{12}+b_{12}}\\{a_{21}+b_{21}}&{a_{22}+b_{22}}\end{pmatrix}}$

${\mbox{Multiplication by multiplying scalar c to all entries by M.}}$

$c\cdot M_{2\times 2}\ {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}={\begin{pmatrix}c\cdot a_{11}&c\cdot a_{12}\\c\cdot a_{21}&c\cdot a_{22}\end{pmatrix}}$

${\mbox{Zero matrix has all entries = 0:}}$

$0_{M_{m\times n}}={\begin{pmatrix}0&\cdots &0\\\vdots &&\vdots \\0&\cdots &0\end{pmatrix}}$
Ex.3. $\mathbb {C}$ form a vector space over $\mathbb {R}$ .
Ex.4. ${\mbox{F is a vector space over itself.}}$
Ex.5. $\mathbb {R}$ is a vector space over $\mathbb {Q}$ .
Ex.6. ${\mbox{Let S be a set. Let}}$
${\mathcal {F}}(S,\mathbb {R} )={\big \{}f:S\to \mathbb {R} {\big \}}$
$f,g\in {\mathcal {F}}(S,\mathbb {R} )$
$(f+g)(t)=f(t)+g(t)\ for\ any\ t\in S$
$(af)(t)=a\cdot f(t)\$

@@ Line 13: / Line 13: @@
 ==Force Vectors==
 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''====
-<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>
-#<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> 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=====
@@ 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
-: <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> \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> 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>
 -----
-. <math> x \mapsto \vert x\vert \in \mathbb{R} \ \vert x y\vert \le \vert x\vert \vert y\vert </math>
+. <math> x \mapsto \vert x\vert \in \mathbb{R} \ \vert x+y\vert \le \vert x\vert+\vert y\vert </math>
 ====''Examples''====
 '''Ex.1.'''
@@ Line 50: / Line 50: @@
 <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 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> 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/>
 '''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>

06-240/Classnotes For Thursday, September 21: Difference between revisions

Latest revision as of 06:42, 11 July 2007

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