How do i find basis of this vector?

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mathsRus
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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!
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username1732133
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(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.
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DFranklin
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(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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