数学 - 線形代数学 - ベクトル空間 – 和と直和

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)、5(和と直和)、練習問題1、2、3.を取り組んでみる。

$\begin{array}{l} v \in V, u, u' \in U, w, w' \in W \\ v = u + w \\ v = u' + w' \\ v = (x, y) \\ u = a_{1} (0, 1) \\ u' = a_{2} (0, 1) \\ w = a_{3} (2, 1) \\ w' = a_{3} (2, 1) \\ (x, y) = a_{1} (0, 1) + a_{3} (2, 1) \\ (x, y) = a_{2} (0, 1) + a_{4} (2, 1) \\ x = 2 a_{3} \\ a_{3} = \frac{x}{2} \\ y = a_{1} + a_{3} \\ a_{1} = y - \frac{x}{2} \\ x = 2 a_{4} \\ a_{4} = \frac{x}{2} \\ y = a_{2} + a_{4} \\ a_{2} = y - \frac{x}{2} \\ u = u', v = v' \\ (x, y) = a_{1} (2, 1) + a_{2} (1, 1) \\ x = 2 a_{1} + a_{2} \\ a_{2} = x - 2 a_{1} \\ y = a_{1} + a_{2} \\ y = a_{1} + x - 2 a_{1} \\ a_{1} = x - y \\ a_{2} = x - 2 x + 2 y = - x + 2 y \end{array}$
$\begin{array}{l} (x, y, z) = a_{1} (1, 0, 0) + (a_{2} (1, 1, 0) + a_{3} (0, 1, 1)) \\ x = a_{1} + a_{2} \\ y = a_{2} + a_{3} \\ z = a_{3} \\ y = a_{2} + z \\ a_{2} = y - z \\ x = a_{1} + y - z \\ a_{1} = x - y + z \\ w = (x - y + z) (1, 0, 0) \\ u = (y - z) (1, 1, 0) + z (0, 1, 1) \end{array}$
$\begin{array}{l} A - c B \neq O \\ C = a_{1} A + a_{2} B \end{array}$

Kamimura's blog