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

• April 13th 2010, 02:22 PM
DarK
Determine if W is a subspace of the vector space V.
Could someone help me out with questions (a) and (c), just the first couple of steps.

Any help is appreciated! (Rofl)
• April 13th 2010, 07:06 PM
dwsmith
The axioms are scalar multiplication and vector addition, correct?
• April 13th 2010, 07:13 PM
DarK
Quote:

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
• April 13th 2010, 07:23 PM
dwsmith
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.
• April 13th 2010, 07:34 PM
dwsmith
The key to the addition is that a and b aren't negative.
• April 13th 2010, 08:07 PM
DarK
Quote:

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?
• April 13th 2010, 08:10 PM
dwsmith
If k is negative, a and b are still positive.

ka=w would be negative.
• April 13th 2010, 08:14 PM
dwsmith
The reason the problem says that is so this doesn't happen:

$a+bx^2+a+(-bx^2)=2a \not\in V$
• April 13th 2010, 08:19 PM
DarK
Ok, that makes alot more sense, thank you!