2个子空间交集的维数 交换$\perp$

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Chapter 2 Quadric, Exercise 3.6
5. Prove that if $L ⊆ V$ and $M ⊆ V$ are vector subspaces of the same dimension then,$$\dim(L\cap M^\perp)=\dim(L^\perp\cap M)$$

证明

$$\dim(L\cap M^\perp)=\dim(L)-\dim(L\cap M)=\dim(M)-\dim(L\cap M)=\dim(L^\perp\cap M)$$