- #1
Benny
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Hi can someone assist me with the following question?
Q. Let V be a finite dimensional real vector space with inner product < , > and let W be a subspace of V. Then the orthogonal complement of W is defined as follows.
[tex]
W^o = \{ v \in V: < v,w > = 0,w \in W\}
[/tex]
Prove the following:
a) [tex]W^o[/tex] is a subspace of V.
b) [tex]W \cap W^o = \left\{ {\mathop 0\limits^ \to } \right\}[/tex]
c) [tex]\dim W + \dim W^o = \dim V[/tex]
My working:
I can do the first part but the others are a problem for me.
b) W and W^o are both subspaces of V and so they both contain the zero vector. Then their intersection also contains the zero vector. Suppose the intersection contains some non-zero vector say f. Then we must have <f,f> = 0 for some non-zero vector f. But this contradicts some inner product property which says <f,f> = 0 iff f = zero vector. So from that I conclude that [tex]W \cap W^o = \left\{ {\mathop 0\limits^ \to } \right\}[/tex].
c) I can't think of a way to do this one. I know that dim(V) >= dim(W), dim(W_0) because any linearly independent set in V has most k elements where k is the number of vectors in a basis for V.
Can someone help me with part c or check my answer for part b? Any help appreciated.
Q. Let V be a finite dimensional real vector space with inner product < , > and let W be a subspace of V. Then the orthogonal complement of W is defined as follows.
[tex]
W^o = \{ v \in V: < v,w > = 0,w \in W\}
[/tex]
Prove the following:
a) [tex]W^o[/tex] is a subspace of V.
b) [tex]W \cap W^o = \left\{ {\mathop 0\limits^ \to } \right\}[/tex]
c) [tex]\dim W + \dim W^o = \dim V[/tex]
My working:
I can do the first part but the others are a problem for me.
b) W and W^o are both subspaces of V and so they both contain the zero vector. Then their intersection also contains the zero vector. Suppose the intersection contains some non-zero vector say f. Then we must have <f,f> = 0 for some non-zero vector f. But this contradicts some inner product property which says <f,f> = 0 iff f = zero vector. So from that I conclude that [tex]W \cap W^o = \left\{ {\mathop 0\limits^ \to } \right\}[/tex].
c) I can't think of a way to do this one. I know that dim(V) >= dim(W), dim(W_0) because any linearly independent set in V has most k elements where k is the number of vectors in a basis for V.
Can someone help me with part c or check my answer for part b? Any help appreciated.