I seem to have a good understanding of spanning sets and linear independence which then becomes essential for understanding basis, but I am unsure how all this works for the field of polynomials.
I know that P${_3}$($\mathbb{R}$) is the set of all polynomials degree less than or equal to 3. Therefore it has a standard basis of $$\{1,x,x^2,x^3\}$$
What I am having trouble with is understanding how to create a basis from a spanning set.
In my case the spanning set is $$S = \{ 1+2x , 1+x+x^2 , 2+x-x^2 , 3+2x , x-2x^3 \}$$
Because the standard basis has $x^3$ as an element, does this mean that $x-2x^3$ has to automatically be an element of the basis or am I going about this all wrong?
Any help is greatly appreciated.
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$\begingroup$It is as you have said, you know that $S$ is a subspace of $P_3(\mathbb{R})$ (and may even be equal) and the dimension of $P_3(\mathbb{R}) = 4$. You know the only way to get to $x^3$ is from the last vector of the set, thus by default it is already linearly independent. Find the linear dependence in the rest of them and reduce the set to a linearly independent set, thus its own basis!
One way to find the linear independence might be to write them as column vectors in a matrix $A$ and row reduce.
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