Since , it can not be a spanning set. There are only two independent functions and the subspace has 3 dimensions.
- Hollywood
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 .
Hollywood's answer is best. Being not as sharp, I would have gone directly to the definition of "span":
Given any numbers, , , and , do there exist numbers, a, b, and c, such that
So , , and
From , .
From .
Putting those into the first equation, .
Since the "a"s cancelled we cannot solve for a.
In fact, in order that there exist such a representation of we must have so that . That is, , , and span the two dimensional subspace having as basis.