06-240/Classnotes For Thursday, September 21

A force has a direction & a magnitude.

Force Vectors

1. [math]\displaystyle{ \mbox{There is a special force vector called 0.} }[/math]
2. [math]\displaystyle{ \mbox{They can be added.} }[/math]
3. [math]\displaystyle{ \mbox{They can be multiplied by any scalar.} }[/math]

Properties

[math]\displaystyle{ \mbox{(convention: }x,y,z \mbox{ are vectors; }a,b,c \mbox{ are scalars)} }[/math]

1. [math]\displaystyle{ x+y=y+x \ }[/math]
2. [math]\displaystyle{ x+(y+z)=(x+y)+z \ }[/math]
3. [math]\displaystyle{ x+0=x \ }[/math]
4. [math]\displaystyle{ \forall x\; \exists\ y \ \mbox{ s.t. }x+y=0 }[/math]
5. [math]\displaystyle{ 1\cdot x=x \ }[/math]
6. [math]\displaystyle{ a(bx)=(ab)x \ }[/math]
7. [math]\displaystyle{ a(x+y)=ax+ay \ }[/math]
8. [math]\displaystyle{ (a+b)x=ax+bx \ }[/math]

Definition

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

[math]\displaystyle{ +: V \times V \to V }[/math]

[math]\displaystyle{ \cdot: F \times V \to V \mbox{, so that:} }[/math]

1. [math]\displaystyle{ \forall x,y \in V\ x+y=y+x }[/math]
2. [math]\displaystyle{ \forall x,y \in V\ x+(y+z)=(x+y)+z }[/math]
3. [math]\displaystyle{ \exists\ 0 \in V s.t.\; \forall x \in V\ x+0=x }[/math]
4. [math]\displaystyle{ \forall x \in V\; \exists\ y \in V\ s.t. \ x+y=0 }[/math]
5. [math]\displaystyle{ 1\cdot x=x\ }[/math]
6. [math]\displaystyle{ a(bx)=(ab)x\ }[/math]
7. [math]\displaystyle{ a(x+y)=ax+ay\ }[/math]
8. [math]\displaystyle{ \forall x \in V\ ,\forall a,b \in F\ (a+b)x=ax+bx }[/math]

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9. [math]\displaystyle{ x \mapsto \vert x\vert \in \mathbb{R} \ \vert x+y\vert \le \vert x\vert+\vert y\vert }[/math]

Examples

Ex.1. [math]\displaystyle{ F^n= \lbrace(a_1,a_2,a_3,\ldots,a_{n-1},a_n):\forall i\ a_i \in F \rbrace }[/math]
[math]\displaystyle{ n \in \mathbb{Z}\ , n \ge 0 }[/math]
[math]\displaystyle{ x=(a_1,\ldots,a_2)\ y=(b_1,\ldots, b_2)\ }[/math]
[math]\displaystyle{ x+y:=(a_1+b_1,a_2+b_2,\ldots,a_n+b_n)\ }[/math]
[math]\displaystyle{ 0_{F^n}=(0,\ldots,0) }[/math]
[math]\displaystyle{ a\in F\ ax=(aa_1,aa_2,\ldots,aa_n) }[/math]
[math]\displaystyle{ \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]
[math]\displaystyle{ 7\left( \frac{1}{5},\frac{1}{7},\frac{1}{9}\right)=\left( \frac{7}{5},1,\frac{7}{9}\right) }[/math]
Ex.2. [math]\displaystyle{ 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]
[math]\displaystyle{ M_{3\times 2}( \mathbb{R})\ni \begin{pmatrix} 7 & -7 \\ \pi & \mathit{e} \\ -5 & 2 \end{pmatrix} }[/math]
[math]\displaystyle{ \mbox{Addition by adding entry by entry:} }[/math]

[math]\displaystyle{ 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]

[math]\displaystyle{ \mbox{Multiplication by multiplying scalar c to all entries by M.} }[/math]

[math]\displaystyle{ 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]

[math]\displaystyle{ \mbox{Zero matrix has all entries = 0:} }[/math]

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