
spanning set V
Hello,
How should I solve this question.thanks.
Let X = {f,g,h} where
f(x) = 3 + x, g(x) = 1 + x^5 and h(x) = 1 + x 2x^5
Let V ={f is an element of F such that f(x)=α+βx+γx^5
a)determine whether X is a spanning set for V.
(a) Determine whether X is a spanning set for V .

Re: spanning set V
Since $\displaystyle h(x)+2g(x) = 1+x2x^5+2+2x^5 = 3+x = f(x)$, it can not be a spanning set. There are only two independent functions and the subspace has 3 dimensions.
 Hollywood

Re: spanning set V
Hollywood's answer is best. Being not as sharp, I would have gone directly to the definition of "span":
Given any numbers, $\displaystyle \alpha$, $\displaystyle \beta$, and $\displaystyle \gamma$, do there exist numbers, a, b, and c, such that
$\displaystyle a(3+ x)+ b(1+ x^5)+ c(1+ x 2x^5)= \alpha+ \beta x+ \gamma x^5$
$\displaystyle (3a+ b+ c)+ (a+ c)x+ (b 2c)x^5= \alpha+ \beta x+ \gama x^5$
So $\displaystyle 3a+ b+ c= \alpha$, $\displaystyle a+ c= \beta$, and $\displaystyle b 2c= \gamma$
From $\displaystyle a+ c= \beta$, $\displaystyle c= \beta a$.
From $\displaystyle b= \gamma+ 2c= \gamma+ 2\beta 2a$.
Putting those into the first equation, $\displaystyle 3a+ b+ c= 3a+ \gamm+ 2\beta 2a+ \beta a= \gamma+ 3\beta= \alpha$.
Since the "a"s cancelled we cannot solve for a.
In fact, in order that there exist such a representation of $\displaystyle \alpha + \beta x+ \gamma x^5$ we must have $\displaystyle \alpha= \gamma+ 3\beta$ so that $\displaystyle \alpha+ \beta x+ \gamma x^5= \gamma+ 3\beta+ \beta x+ \gamma x^5= (3+ x)\beta+ (1+ x^5)\gamma$. That is, $\displaystyle 3+x$, $\displaystyle 1+ x^5$, and $\displaystyle 1+ x 2x^5$ span the two dimensional subspace having $\displaystyle \{3+ x, 1+ x^5\}$ as basis.