Spanning sets of polynomial functions

[karnten07]

[SOLVED] Spanning sets of polynomial functions (urgent)

Homework Statement

Which o the following are spanning sets for the vector space P2 of polynomial functions of degree $$\leq$$2? (give reasons or your answers)

a.) 2, t^2, t, 2t^2 +3

b.)t+2, t^2 -1

The Attempt at a Solution

I'm not entirely sure how to do this but i think i need to show that any vector in P2 can be written as a linear combination of the elements in the set.

So for a.) i wrote out the generic polynomial of P2 with scalar coefficients:

at^2 + bt + c= 2p + qt^2 + xt + 2yt^2 + 3z

In this case a = q+2y, b = x, c = 2p+3z

I then put this into an augmented matirx and reduced it to:

p 0 0 0 3z/a c/a
0 q 0 2y 0 a
0 0 x 0 0 b

I don't really know what i am supposed to do from here, i was just following a worked example but the conclusion is unclear in it and i don't knoaw what to do. I followed the same procedure or b.) and got this augmented matrix:

a -b/2 0 z/2
0 0 0 x-z
0 0 0 y+z

Im a bit unsure if this is right, but i don't know how to proceed. Any help please, urgent also.

 

[mighty2000]

Hello,

To determine if a set of vectors is a spanning set for a vector space, you need to show that every vector in the vector space can be written as a linear combination of the vectors in the set. In other words, for every vector v in P2, there exist scalars a, b, c such that v = a(2) + b(t^2) + c(t) + a(2t^2 + 3).

For set a), you correctly showed that any vector v in P2 can be written as a linear combination of the vectors in the set. This means that set a) is a spanning set for P2.

For set b), you also correctly showed that any vector v in P2 can be written as a linear combination of the vectors in the set. This means that set b) is also a spanning set for P2.

Therefore, both set a) and set b) are spanning sets for P2.