# Proving a set is a subspace

• May 5th 2009, 11:11 AM
needsmathhelp101
Proving a set is a subspace
Hey everybody,

My tutor has found an old exam online and is having me complete it as a preparatory exercise before my final exam this week. I'm having trouble even wrapping my head around this one, so I'm hoping someone here can help me through it!

(note: the ^ symbol is supposed to be a perpendicular symbol)

Let W be a subspace of the inner product space V. Define
W^ = {v in V: <v,w> = 0 for all w in W}
(a) Prove that W^ is a subspace of V .

If you'd like the test itself, its found at this address, problem number 12.a:

http://www.math.niu.edu/courses/math...Fa04-final.pdf

Thanks so much for any help!
• May 5th 2009, 11:25 AM
HallsofIvy
Quote:

Originally Posted by needsmathhelp101
Hey everybody,

My tutor has found an old exam online and is having me complete it as a preparatory exercise before my final exam this week. I'm having trouble even wrapping my head around this one, so I'm hoping someone here can help me through it!

(note: the ^ symbol is supposed to be a perpendicular symbol)

Let W be a subspace of the inner product space V. Define
W^ = {v in V: <v,w> = 0 for all w in W}
(a) Prove that W^ is a subspace of V .

If you'd like the test itself, its found at this address, problem number 12.a:

http://www.math.niu.edu/courses/math...Fa04-final.pdf

Thanks so much for any help!

The only way I know to prove something is a "subspace" is to use the definition of "subspace"! And that is "U is a subspace of V if and only if
1) U is a subset of V
2) U is closed under addition (the sum of two vectors in U is in U).
3) U is closed under scalar multiplication (a number times a vector in U is in U).

W^ is clearly a subset of V because the definition of W^ begins "{v in V...".

Suppose u and v are both in W^. Then <u, w>= 0 and <v, w>= 0 for all w i W. What is <u+ v, w>?

Suppose a is a scalar (number) and v is in W^. Then <v, w>= 0 for all w in W. What is <au, w>?