Given $=\{2+,1−,^3,^4+^5,^3−2^5,^2\}$ Prove that $()=\Bbb R_5[]$.
How do I give a definite proof that a given span is equal to the entire space of polynomials $\Bbb R_5[]$?
Proving this span is a given space of polynomials
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linear-algebra
2 Answers
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All of the polynomials is $A$ are polynomials of degree at most five. Also $Sp(A)$ is always a subspace. So therefore you have that $Sp(A)$ is a subspace of $R_5[x]$. To show that they are equal it suffices to show that they have the same dimension (ie 6-dimensional). Since $A$ contains 6 elements you just need to show that they are linearly independent and you're done.
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To prove that a given set of vectors span the whole space $V$, it suffices to show that for some known spanning set $S$ (for instance a basis) of $V$, every element of $S$ is a linear combination of the given vectors. This is because by definition of a spanning set, every vector subspace that contains each of them contains the whole space they span. Now there are various candidates for spanning sets of the space of polynomials of degree${}<6$; the most obvious one is that of the monomials $1,x,x^2,\ldots,x^5$, but a bit of easy theory that a set containing a polynomial of each degree (here $0,1,\ldots,5$) will be spanning for the space of polynomials $\Bbb R_5[x]$.
Now your given set already contains polynomials of degrees $1$, $2$, $3$, and $5$. It suffices to construct a linear combination that is of degree$~0$ (a nonzero constant polynomial), and another one of degree$~4$, and your done.