Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
Suppose that $p_0, p_1, ..., p_m$ are polynomials in $P_m(F)$ such that $p_i(2)=0$ for each $i$. Prove that the set of vectors $p_0, ..., p_m$ is linearly dependent.
Note that $P_m(F)$ denotes the vector space of all polynomials of degree less than or equal to $m$.
Suppose that $p_0, p_1, ..., p_m$ are polynomials in $P_m(F)$ such that $p_i(2)=0$ for each $i$. Prove that the set of vectors $p_0, ..., p_m$ is linearly dependent.
If each $p_i(t)$ evaluates to $0$ at $t = 2$, why can you conclude that the $p_i$ must therefore be linearly dependent?:
For intuition: Try to construct linearly independent polynomials, say for $P_2(F)$, such that for each, $p_i(2) = 0$; and see why these polynomials cannot, in fact, be linearly independent.
Suppose there are $m+1$ linearly independent polynomials such that $p_i(2) = 0 \forall p_i$. Then $p_0 = 0$. Hence, e.g., 1 would necessarily be linearly independent of the $m+1$ $p_i$, also in $P_m(F)$. That would give a basis of $P_m(F)$ that has $m+2$ linear independent vectors. Impossible!
If $p_0,\dots,p_m$ were linearly independent and $p_i(2)=0$ for each $i$, then also $p_0,\dots,p_m,1$ would be linearly independent (see below for details). Hence you would have a base of $P_m$ with $m+2$ elements...
some more detail
Suppose: $$ \lambda_0 p_0 + \dots + \lambda_m p_m + \lambda_{m+1} 1 = 0 $$ then evaluating each polinomial in $t=2$ we get $$ 0 + \dots + 0 + \lambda_{m+1} = 0 $$ hence $\lambda_{m+1}=0$. So $$ \lambda_0 p_0 + \dots + \lambda_m p_m = 0 $$ which, by contradiction, we suppose that implies $\lambda_0 = 0, \dots, \lambda_m = 0$. Hence if $p_0,\dots,p_m$ are linearly independent also $p_0, \dots, p_m, 1$ would be linearly independent.
