Proving a Vector Space

• Oct 29th 2007, 05:25 PM
Proving a Vector Space
I just can't completely answer this question, does it want me to prove it using all 10 axioms? Or is there another way

http://i89.photobucket.com/albums/k2...atrixHelp2.jpg

Thanks
• Oct 29th 2007, 07:52 PM
ThePerfectHacker
Here is an easier way to do this without using all 10 properties of a vector space.

Theorem: Let $\displaystyle V$ be a vector space over a field (you probably learned it over $\displaystyle \mathbb{R}$). Let $\displaystyle W$ be a subset of $\displaystyle V$ so that $\displaystyle \bold{a}+\bold{b} \in W$ for all $\displaystyle \bold{a},\bold{b} \in W$ and $\displaystyle c\bold{a} \in W$ for all $\displaystyle \bold{a}\in W$ and $\displaystyle c\in \mathbb{R}$. Then $\displaystyle W$ is a vector space over the field ($\displaystyle \mathbb{R}$).

So you only need to check closure of vector addition and scalar multiplication.
• Oct 29th 2007, 08:00 PM
I'm not sure if I'm correct but, the closure under addition wouldn't be true because the final matrix has to end up as

[a+c 2]
[ 2 b+d]
???

I'm almost positive axiom 6 for scalar multiplication holds, but I'm not sure about axiom 1, if you could tell me whether my above theory is correct or not that would be great. I am not very good at proving axioms at least 1 and 6, because there is no left and right hand side of the equation to compare, so I get really lost.
• Oct 30th 2007, 12:32 AM
kalagota
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