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Let $F$ be a field. $V$ is a vector space over $F$, consisting of all polynomias of degree less or equal to $3$ with coefficients in $F$. Does the sequence $S:\,x,(1-x^2),x^3$ span $V$ ?

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1 Answer 1

Note the dimension of $V$ is 4 and $|S|=3$ so it's impossible for $S$ to span $V$.

Indeed, you cannot generate constant polynomials using this basis.

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And the stadart basis for V is S:1,x,x^2,x^3, right? –  user2013804 Oct 22 '13 at 15:53
    
@user2013804 Yes, the standard basis for $V$ would be $\{1, x, x^2, x^3\}$. But any set of $4$ polynomials with different degree would do -- e.g. $\{5, 3-x, x^2-4x+7, x^2-x^3\}$. –  gt6989b Oct 22 '13 at 17:28

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