Vector subspaces

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January 11th 2010, 08:09 AM
Matharch

Vector subspaces

I am having difficulty with vector subspaces. It would be helpful if someone could give me an answer to at least one of these two questions.

1.

Prove that $W_1= \{(a_1,a_2· \cdot \cdot \cdot ,a_n) \in F^n : a_1+a_2\ + \cdot \cdot \cdot a_n =0 \}$ is a subspace of F^n, but $W_2= \{(a_1,a_2\,\cdot \cdot \cdot a_n) \in F^n : a_1+a_2 + \cdot \cdot \cdot a_n =1 \}$ is not

2.

Is the set W = { f(x) $\in$ P(F): f(x)=0 or f(x) has a degree n} a subspace of P(F) if n ≥ 1? Justify your answer
January 11th 2010, 11:51 AM
abender

$W_1= \{(a_1,a_2,...,a_n) \in F^n : a_1+a_2+...+ a_n =0 \}$

If $a=(a_1,a_2,...,a_n)\in W_1$ , $b=(b_1,b_2,...,b_n) \in W_1$ and $t,u \in \mathbb{F}$ then

$ (ta+ub)_1 + (ta+ub)_2 + ... + (ta+ub)_n = (ta_1+ub_1)+(ta_2+ub_2)+...+ (ta_n+ub_n)$

$= t(a_1+a_2+...+a_n) + u(b_1+b_2+...+b_n)$

$ = t\cdot0+u\cdot 0 $

$ = 0 $

Therefore, $ta+ub \in W_1 \implies W1$ is a subspace.
January 11th 2010, 11:56 AM
abender

$ W_2= \{(a_1,a_2,...,a_n) \in F^n : a_1+a_2 +...+a_n =1 \} $

$W_2$ is not a vector space because $(0,0,...,0) \not\in W_2$ . Note that $(0,0,...,0)$ is simply the zero vector.
January 12th 2010, 05:41 AM
HallsofIvy

Quote:

Originally Posted by Matharch

2.[/COLOR]

Is the set W = { f(x) $\in$ P(F): f(x)=0 or f(x) has a degree n} a subspace of P(F) if n ≥ 1? Justify your answer[/COLOR]

[/tex]f(x)= x^2+ 1[/tex] is a polynomial of degree 2. $g(x)= -x^2+ 3$ is a polynomial of degree 2 also. Is there sum a polynomial of degree 2? Is the set of "polynomials of degree 2" closed under addition?
January 12th 2010, 07:53 AM
Matharch

Thanks guys! It's so simple, I'm almost ashamed that I posted here.
January 12th 2010, 12:52 PM
Raoh

Quote:

Originally Posted by HallsofIvy

[/tex]f(x)= x^2+ 1[/tex] is a polynomial of degree 2. $g(x)= -x^2+ 3$ is a polynomial of degree 2 also. Is there sum a polynomial of degree 2? Is the set of "polynomials of degree 2" closed under addition?

i didn't think that the second question could be answered this way(Doh).
the "Or" made me confused,since it means union.
(Happy)

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