Question

This is a question for Advanced Linear Algebra. Please answer ALL parts well and completely, and please write CLEARLY, with neat, non-cursive writing. Make sure x's and v's and y's, s's and such are clearly different. I sometimes have trouble reading the difference with people's handwriting. Thank you, thumbs up if you can!

(5) §5.11. Suppose X,y are subspaces of an inner-product space V. Show directly (do not use the fact (X1)1x) that (XnX+

Answer 1

Let be subspaces of

$(X\cap Y)^{\perp}=X^{\perp}+Y^{\perp}$

For ,

Let

$x\in (X\cap Y)^{\perp}\\ \Rightarrow \\ x.a=0,a\in X\cap Y\\ x.a=0,a\in X\&a\in Y\\ x.2a=x.(a+a)=x.a+x.a,a\in X,a\in Y\\ x\in X^{\perp}+Y^{\perp}\\ (X\cap Y)^{\perp}\subseteq X^{\perp}+Y^{\perp}--(1)$

Let

now let 0 E X1 yl

By(1),(2),

$X^{\perp}+Y^{\perp}= (X\cap Y)^{\perp}$