Can anyone help me with this elementary problem? How do I prove the following is a subspace?

W = {(a1, a2, a3) | 2*a1 - 7*a2 +a3 = 0}

I know I have to prove its closed under addition and multiplication... But the condition on W is confusing me...

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- Jun 1st 2009, 06:17 PMpaupsersProving this is a subspace
Can anyone help me with this elementary problem? How do I prove the following is a subspace?

W = {(a1, a2, a3) | 2*a1 - 7*a2 +a3 = 0}

I know I have to prove its closed under addition and multiplication... But the condition on W is confusing me... - Jun 1st 2009, 06:26 PMpaupsers
I think I just solved it, actually!

I let a3 = 7*a2 - 2*a1, and made that the new condition on W.

I then let a = (a1, a2, 7*a2 - 2*a1) and similarly for b.

Then I did a + cb = (a1 + c*b1, a2 + c*b2, 7(a2 + c*b2) - 2(a1 + c*b1))

Which satisfies my new condition on W I believe...

Can anyone verify this? - Jun 1st 2009, 06:36 PMRandom Variable
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Let and be in

then which is in

I'm sure you can then prove that it's also closed under scalar multiplication. - Jun 1st 2009, 06:42 PMRandom Variable
If you want to prove both at the same time, you have to show that is in

- Jun 1st 2009, 07:37 PMmath2009

Obviously, is linear transformation

Finally, W is a subspace. - Jun 1st 2009, 07:47 PMIsomorphism