09-240/Classnotes for Thursday September 24

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WARNING: The notes below, written for students and by students, are provided "as is", with absolutely no warranty. They can not be assumed to be complete, correct, reliable or relevant. If you don't like them, don't read them. It is a bad idea to stop taking your own notes thinking that these notes can be a total replacement - there's nothing like one's own handwriting! Visit this pages' history tab to see who added what and when.
Convention for today:
x, y, z \in \mathbf V
a, b, c \in F

A V.S. over F: V, 0, +, \cdot s.t.

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

Examples

  1. \left\{ \begin{pmatrix} a_1 \\ \vdots \\ a_n \end{pmatrix} : a_i \in F \right\} = F^n
  2. \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)
  3. Let S be a set (F is some field)
    \mathcal F(S, F) = \{f: S \rightarrow F\}
    S = Primary colours = {red, green, blue}
    F = F2 = {0, 1}
    \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\}

S = \mathbb N = \{ 1, 2, 3, 4, \ldots \} F = \mathbb R

\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} \}

S = \begin{pmatrix} \vdots \\ \vdots \end{pmatrix} n \Rightarrow \mathcal F(S, F) = F^n

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

Claim: + is associative. Given f, g, h \in \mathcal F(S, F), (f + g) + h = f + (g + h) \forall \sigma

((f + g) + h)(\sigma) = (f + g)(\sigma) + h(\sigma)
= (f(\sigma) + g(\sigma)) + h(\sigma)
= f(\sigma) + g(\sigma) + h(\sigma) \mbox{ (by F2)}
(f + (g + h))(\sigma) = f(\sigma) + (g + h)(\sigma)
= f(\sigma) + (g(\sigma) + h(\sigma))
= f(\sigma) + g(\sigma) + h(\sigma)
  1. \mathbb C \mbox{ is a V.S. over } \mathbb R
  2. \mathbb R \mbox{ is a V.S. over } \mathbb Q
  3. \mathbb R \mbox{ is a V.S. over } \mathbb R
  4. \,\! \{0\} \mbox{ is a V.S. over } F

Dull theorem

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