[微積] Marsden 高微 第一章第20題 關於orthogonal正交

[t0444564 (艾利歐)]

題目是:
Let S and T be nonzero orthogonal subspaces of R^n. Prove that if S and T are
orthogonal complements (that is, S and T span all of R^n), then S交集T={0}
and dim(S) + dim(T) = n , where dim(S) denotes dimension of S. Give examples
in R^3 of nonzero orthogonal subspaces for which the condition dim(S) + dim(T)
= n holds and examples where it fails. Can it fail in R^2?

打的有點冗長,而且有學過一點線性代數就會覺得太trivial.
結果完全不知道怎麼證(或說我可以使用甚麼事實去證?)

至於Examples , 我看不太懂他想要我給甚麼樣的例子(應該說,不懂題意.)
甚麼是失敗在R^2?(是說正交的情況失敗在R^2嗎?,那只有S和T同方向才失敗吧?!)

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→ mzhrqoc01   :例子應該是要說明dim(S) + dim(T) = n必須要有S and
→ mzhrqoc01   :T span all of R^n的條件，但在R^2這個條件自然成立