06-240/Classnotes For Thursday, September 21

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Scan of Lecture Notes

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Scan of Tutorial notes

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Force Vectors

A force has a direction and a magnitude.

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{ }\mbox{ are vectors; }a,b,c\mbox{ }\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} }[/math]