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**x3bnm** We know the definition of span:

If $\displaystyle S = \{v_1, v_2,....,v_r\}$ is a set of vectors in a vector space $\displaystyle V$, then the subspace $\displaystyle W$ of $\displaystyle V$ consisting of all linear combinations of the vectors in $\displaystyle S$ is called the space spanned by $\displaystyle v_1, v_2,....,v_r$, and we say that the vectors $\displaystyle v_1, v_2,....,v_r$ span $\displaystyle W$. To indicate that $\displaystyle W$ is the space spanned by the vectors in the set $\displaystyle S = \{v_1, v_2,....,v_r\}$, we write:

$\displaystyle W = span(S)$ or $\displaystyle W = span\{v_1, v_2, ....., v_r\}$

I've a math problem and asked to find out whether the polynomials span a space or not. Determine whether the following polynomials span $\displaystyle P_2$ (the set of degree 2 polynomials).

$\displaystyle

p_1 = 1 - x + 2.x^2, p_2 = 3 + x, p_3 = 5 - x + 4x^2, p_4 = -2 - 2x + 2x^2

$

The book says it doesn't span $\displaystyle P_2$

How do they figure out that $\displaystyle P_2$ subspace does not consist of all linear combination of polynomials $\displaystyle v_1, v_2,....,v_r$? How does one come to this conclusion?