数学 - 線形代数学 - 線形写像 – 線形写像(線分、凸集合)

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4章(線形写像)、2(線形写像)、練習問題10-19.を取り組んでみる。

$\begin{array}{l} 0 \leq t \leq 1 \\ L (t P + (1 - t) Q) \\ = t L (P) + (1 - t) L (Q) \end{array}$
$\begin{array}{l} P, Q \in S \\ 0 \leq t \leq 1 \\ L (t P + (1 - t) Q) \\ = t L (P) + (1 - t) L (Q) \\ \geq t 0 + (1 - t) 0 \\ \geq 0 \\ t P + (1 - t) Q \in S \end{array}$
$\begin{array}{l} P, Q \in S \\ 0 \leq t \leq 1 \\ L (t P + (1 - t) Q) \\ = t L (P) + (1 - t) L (Q) \\ > t c + (1 - t) c \\ = c \end{array}$
$\begin{array}{l} P, Q \in S_{1} \cap S_{2} = S \\ 0 \leq t \leq 1 \\ t P + (1 - t) Q \in S_{1} \\ t P + (1 - t) Q \in S_{2} \\ t P + (1 - t) Q \in S_{1} \cap S_{2} = S \end{array}$
$\begin{array}{l} P, Q \in S' \\ 0 \leq t \leq 1 \\ L (t P + (1 - t) Q) \\ = t L (P) + (1 - t) L (Q) \\ L (P), L (Q) \in S \\ t L (P) + (1 - t) L (Q) \in S \\ t P + (1 - t) Q \in S' \end{array}$
$\begin{array}{l} 0 \leq t \leq 1 \\ (t X_{1} + (1 - t) X_{2}) \cdot A \\ = t X_{1} \cdot A + (1 - t) X_{2} \cdot A \\ \geq t c + (1 - t) c \\ = c \end{array}$

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