I am having a little trouble with the following questions.
Q: Are the following sets vector spaces with the indicated operations
a) The set of all polynomials of degree <=3 ; the operations of P
b) The set V of 2x2 matrices with zero determinant, usual matrix operations
c)The set V of all 2x2 matrices whose entries sum to zero; operations of M22
For part (a), I understand what polynomials with degree <= 3 means but I am not sure what the operations of P are supposed to be.
For part (b) I don't know what it means when a matrix has a determinant of zero but I do know what the usual matrix operations are
For part (c) I don't know what the operations of M22 are
For a subspace all the following three properties must be satisfied:
$1) \ \vec{0} \in W\\ 2) \ \vec{v}+\vec{w} \in W\\ 3) \ \vec{cv}\to c \cdot \vec{v} \ ,c \in \mathbb{R}$
You have to check this properties and show that
For a)
The properties hold, indeed
1) Trivial
2) $(ax^3+bx^2+cx+d) +(ex^3+fx^2+gx+h) = (a+e)x^3+(b+f)x^2+(c+g)x+(d+h)$
3) $k(ax^3+bx^2+cx+d)=kax^3+kbx^2+kcx+kd$
For b) not, indeed
$\begin{pmatrix} 1 & 0 \\0 & 0\end{pmatrix}+\begin{pmatrix} 0 &0 \\ 0 &1 \end{pmatrix}=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
For c) yes. 1) Trivial
2) Let consider
$M=\begin{pmatrix} a & b \\c & d\end{pmatrix}$ such that $a+b+c+d=0$
$N=\begin{pmatrix} e & f \\g & h\end{pmatrix}$ such that $e+f+g+h=0$
then
$M+N=\begin{pmatrix} a+e & b+f \\c+g & d+h\end{pmatrix}$ and $(a+e)+(b+f)+(c+g)+(d+h)=(a+b+c+d)+(e+f+g+h)=0$
3) $kM=\begin{pmatrix} ka & kb \\kc & kd\end{pmatrix}$ and $ka+kb+kc+kd=k(a+b+c+d)=0$