Determine if W is a subspace of the vector space V.

**DarK** · April 13th 2010, 01:22 PM

Could someone help me out with questions (a) and (c), just the first couple of steps.

Any help is appreciated!

**dwsmith** · April 13th 2010, 06:06 PM

The axioms are scalar multiplication and vector addition, correct?

**DarK** · April 13th 2010, 06:13 PM

Originally Posted by dwsmith

The axioms are scalar multiplication and vector addition, correct?

Yes, let me write out the definitions:

V is the vector space

addition: If u & v are in V, then u + v is in V

multiplication: If k is any scalar & u is any object in V, then ku is in V

**dwsmith** · April 13th 2010, 06:23 PM

part c
$\alpha p(x)=\alpha(a+bx^2)=\alpha a + \alpha bx^2$ where $\alpha \in \mathbb{R}$

Therefore, $\alpha a + \alpha bx^2 \in V$

Do addition the same way.

**dwsmith** · April 13th 2010, 06:34 PM

The key to the addition is that a and b aren't negative.

**DarK** · April 13th 2010, 07:07 PM

Originally Posted by dwsmith

The key to the addition is that a and b aren't negative.

What about the multiplication, if I had something like:

k(a+bx^2) = ka + kbx^2

The k could be a negative number, making a and b negative, so the multiplication axiom wouldn't hold?

**dwsmith** · April 13th 2010, 07:10 PM

If k is negative, a and b are still positive.

ka=w would be negative.

**dwsmith** · April 13th 2010, 07:14 PM

The reason the problem says that is so this doesn't happen:

$a+bx^2+a+(-bx^2)=2a \not\in V$

**DarK** · April 13th 2010, 07:19 PM

Ok, that makes alot more sense, thank you!

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