数学 - 線形代数学 - ベクトル空間 – 定義(性質)

学習環境

ラング線形代数学(上)(S.ラング (著)、芹沢正三 (翻訳)、ちくま学芸文庫)の2章(ベクトル空間)、2(定義)、練習問題1、2、3、4、5.を取り組んでみる。

$\begin{array}{l} V S 3, V S 6, V S 8 \\ O = v + (- 1) v \\ = 1 \cdot v + (- 1) v \\ = (1 - 1) v \\ = 0 v \end{array}$
$\begin{array}{l} u \in V \\ u + v \\ = 1 u + 1 v \\ = (\frac{1}{c} c) u + (\frac{1}{c} c) v \\ = \frac{1}{c} (c u) + \frac{1}{c} (c v) \\ = \frac{1}{c} (c u + c v) \\ = \frac{1}{c} (c u + O) \\ = \frac{1}{c} (c u) \\ = (\frac{1}{c} c) u \\ = 1 u \\ = u \\ u + v = u \\ v + u \\ = 1 v + 1 u \\ = (\frac{1}{c} c) v + (\frac{1}{c} c) u \\ = \frac{1}{c} (c v) + \frac{1}{c} (c u) \\ = \frac{1}{c} (c v + c u) \\ = \frac{1}{c} (O + c u) \\ = \frac{1}{c} (c u) \\ = (\frac{1}{c} c) u \\ = 1 u \\ = u \\ v + u = u \\ v = O \end{array}$
$\forall x \in S [f (x) = 0]$
$\begin{array}{l} - v + (v + w) = (v + w) - v = O \\ (- v + v) + w = O - v \\ O + w = - v \\ w = - v \end{array}$
$\begin{array}{l} u \in V \\ u + w \\ = u + w + v - v \\ = u + v - v \\ = u \\ w = O \end{array}$

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