Could someone help me out with questions (a) and (c), just the first couple of steps. Any help is appreciated!
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The axioms are scalar multiplication and vector addition, correct?
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
part c $\displaystyle \alpha p(x)=\alpha(a+bx^2)=\alpha a + \alpha bx^2$ where $\displaystyle \alpha \in \mathbb{R}$ Therefore, $\displaystyle \alpha a + \alpha bx^2 \in V$ Do addition the same way.
The key to the addition is that a and b aren't negative.
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?
Last edited by DarK; Oct 21st 2010 at 07:02 PM.
If k is negative, a and b are still positive. ka=w would be negative.
The reason the problem says that is so this doesn't happen: $\displaystyle a+bx^2+a+(-bx^2)=2a \not\in V$
Ok, that makes alot more sense, thank you!
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