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How do i find basis of this vector?

Hello,

as the title states:

{1 + 2x + x^2, 1 - x- 4x^2, x - x^2, 1 + 3x} in P2

Now i know it does not span as dimP2 = 3 and there are four vectors (I could be wrong). For a basis, do you check for independence?

Thanks!
Original post by mathsRus
Hello,

as the title states:

{1 + 2x + x^2, 1 - x- 4x^2, x - x^2, 1 + 3x} in P2

Now i know it does not span as dimP2 = 3 and there are four vectors (I could be wrong). For a basis, do you check for independence?

Thanks!


What is the actual question that you are trying to answer?

First of all, a vector does not have a basis. A vector space has a basis.

Secondly having "too many vectors" does not prevent a set of vectors from being a spanning set.

Third, yes, a basis is a linearly independent spanning set.

I think that's all correct. Not done any linear algebra for ages.
Original post by BuryMathsTutor
What is the actual question that you are trying to answer?Hope it's OK to piggy back on your answer to emphasize/clarify...

First of all, a vector does not have a basis. A vector space has a basis.
True. From context I think it's clear that he's looking at the vector space of polynomials of degree <=2 and he has a set composed of four elements (therefore vectors in this context) of the vector space.

Secondly having "too many vectors" does not prevent a set of vectors from being a spanning set.

Third, yes, a basis is a linearly independent spanning set.
Both correct.

Note that having "too many vectors" does prevent a set of vectors from being linearly independent - any set with more vectors than the dim of the vector space cannot be a lin indep set.

What's still not very clear is what he's actually supposed to do.

Most likely seems to be "find a subset of this set that forms a basis for P2". There are various ways of doing this, but I would probably proceed as follows (calling the vectors v1, v2, v3, v4).

Start with the empty set as your "basis in progress". Repeatedly:

Take the next vector in the list. Check if it can be written as a lin. comb. of your "basis in progress". If it can, discard it, otherwise add it to the list.

If you can add 3 vectors to the list, you have 3 lin indep vectors over a vector space of dimension 3 and you're done.

If you can't, then your vectors don't span P2 and you can't choose a basis from them.

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