Find a basis of the vector spaces in Problem 1 and evaluate their dimension. Find a basis of the vector spaces below and evaluate their dimension.


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*$\{(x,y,z) \in \textbf{R}^3:x+y+z=0\}$

*$\{(x,y,z) \in \textbf{R}^3:x+y=0$ and $x+z=0\}$

*The space of polynomials $p(x)$ of degree m with real coefficients such that $p(0)=0$.

*The spaces of polynomials $p(x)$ of degree $\leq m$ with complex coefficients such that $p(0)=0$.


Part 1: $(x,y,-x-y)=x(1,0,-1)+y(0,1,-1)$, and this shows that the set is spanned by two linearly independent vectors (1,0,-1) and (0,1,-1).
Part 2: $(x,y,z)=(x,-x,-x) =x(1,-1,-1)$, and this shows that the set is spanned by one linearly independent vector (1,-1,-1).
I have no idea how to have basis for polynomials.

Polynomials are defined by the sequence $a_n$ of their coefficients such that you can write $P(X)=\sum_n a_n X^n$, so a basis is simple $\{1,X,...X^n\}$ of dimension $n+1$.
As noted in the comment, $p(0)=0$ implies that $a_0=0$ and you simply remove $1$ from the basis. Technically, if the polynomial has to be of degree exactly $n$, it's not a vector space any longer as it doesn't contain 0.
For the last question, you need to precise the field you're talking about. The space described is a $n$-dimensional $\mathbb{C}$ vector space but a $2n$-dimensional $\mathbb{R}$ vector space.
