# For these subsets $S$, are they subspace for the indicated vector space $V$

Q1. $V =P_5(R)$ and $S=\{p(x)\mid p(15)=0\}$.

I think it is a subspace, but not 100% sure.
I tried let $p_1(x)=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5$, such that $p_1(15)=0$
$p_2(x)=b_0+b_1x+b_2x^2+b_3x^3+b_4x^4+b_5x^5$, such that $p_2(15)=0$
and proved it's non empty, closed under addition and scalar multpilication.

Q2. $V =F(R,R)$ and $S=\{f(x)\mid f(2)=f(3)\}$.

I think it is a subspace, since it's closed under addition and scalar multiplication

Q3. $V =M_{2×2}(R)$ and $S=\{A\mid {\rm rank}{(A)}\ge1\}$.

I have no idea how to prove it is a subspace and I can't think of any counter example

Thank you

• For Q3, does that set contain the zero vector (=the zero matrix)? Apr 30, 2016 at 11:12
• Questions 1 and 2 are OK. For question 3, $V=\mathcal M_{2\times2}(\mathbf R)^\ast$. Apr 30, 2016 at 11:32

$\sf Q1)$ Indeed, it is closed under addition and scalar multiplication, and it contains the “zero” polynomial, hence it is a subspace.
$\sf Q2)$ Same thing, here the “zero” function satisfies the condition $f(2)=f(3)$, so it's in that set.
$\sf Q3)$ Is the following correct? $$\operatorname{rank}\pmatrix{0&0 \\ 0&0}\geqslant 1.$$ What can you conclude then?