09-240/Classnotes for Thursday September 24

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Convention for today: [math]\displaystyle{ x, y, z \in \mathbf V }[/math] [math]\displaystyle{ a, b, c \in F }[/math]

A V.S. over [math]\displaystyle{ F: V, 0, +, \cdot }[/math] s.t.

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

Examples

1. [math]\displaystyle{ \left\{ \begin{pmatrix} a_1 \\ \vdots \\ a_n \end{pmatrix} : a_i \in F \right\} = F^n }[/math]
2. [math]\displaystyle{ \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]
3. Let S be a set (F is some field)

   [math]\displaystyle{ \mathcal F(S, F) = \{f: S \rightarrow F\} }[/math]

   S = Primary colours = {red, green, blue}

   F = F₂ = {0, 1}

   [math]\displaystyle{ \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]\displaystyle{ S = \mathbb N = \{ 1, 2, 3, 4, \ldots \} F = \mathbb R }[/math]

[math]\displaystyle{ \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]\displaystyle{ S = \begin{pmatrix} \vdots \\ \vdots \end{pmatrix} n \Rightarrow \mathcal F(S, F) = F^n }[/math]

1. [math]\displaystyle{ O_{\mathcal F(S, F)}(\sigma) = 0_F \forall \sigma \in S }[/math]
2. [math]\displaystyle{ f, g \in \mathcal F(S, F) }[/math]
   [math]\displaystyle{ \,\! (f + g)(\sigma) = f(\sigma) + g(\sigma) }[/math]
   [math]\displaystyle{ f \in \mathcal F(S, F) }[/math]
   [math]\displaystyle{ a \in F \Rightarrow \forall \sigma \in S, S(af)(\sigma) = a \cdot (f(\sigma)) }[/math]

Claim: + is associative. Given [math]\displaystyle{ f, g, h \in \mathcal F(S, F), (f + g) + h = f + (g + h) \forall \sigma }[/math]

[math]\displaystyle{ ((f + g) + h)(\sigma) = (f + g)(\sigma) + h(\sigma) }[/math]

[math]\displaystyle{ = (f(\sigma) + g(\sigma)) + h(\sigma) }[/math]

[math]\displaystyle{ = f(\sigma) + g(\sigma) + h(\sigma) \mbox{ (by F2)} }[/math]

[math]\displaystyle{ (f + (g + h))(\sigma) = f(\sigma) + (g + h)(\sigma) }[/math]

[math]\displaystyle{ = f(\sigma) + (g(\sigma) + h(\sigma)) }[/math]

[math]\displaystyle{ = f(\sigma) + g(\sigma) + h(\sigma) }[/math]

4. [math]\displaystyle{ \mathbb C \mbox{ is a V.S. over } \mathbb R }[/math]
5. [math]\displaystyle{ \mathbb R \mbox{ is a V.S. over } \mathbb Q }[/math]
6. [math]\displaystyle{ \mathbb R \mbox{ is a V.S. over } \mathbb R }[/math]
7. [math]\displaystyle{ \,\! \{0\} \mbox{ is a V.S. over } F }[/math]

Dull theorem

1. Cancellation: [math]\displaystyle{ x + y = x + z \Rightarrow y = z }[/math] (add w to both sides s.t. x + w = 0)
2. 0 is unique
3. Negatives are unique: [math]\displaystyle{ x + y = 0 = x + z \Rightarrow y = z }[/math]
4. [math]\displaystyle{ 0x = 0. a \cdot 0 = 0 }[/math]
5. [math]\displaystyle{ (-a) \cdot x = a \cdot (-x) = -(ax) }[/math]