Question

Determine if a. H={(x,y)/ y=3x-1} is a subspace of R2 b. H={at+b/ b=8a} is a subspace of P1

Vectors and spaces
Determine if
$$\displaystyle{a}.{H}={\left\lbrace\frac{{{x},{y}}}{{y}}={3}{x}-{1}\right\rbrace}$$ is a subspace of R2
$$\displaystyle{b}.{H}={\left\lbrace{a}{t}+\frac{{b}}{{b}}={8}{a}\right\rbrace}$$ is a subspace of P1

Let V be a vector space and $$S\subset V$$ is said to be a subspace of V if the followings holds
$$0\in S$$ for u,$$v\in S$$ and for a scaler k $$u+kv\in S$$.
(a)Here it is given $$\displaystyle{H}={\left\lbrace{\left({x},{y}\right)}:{y}={3}{x}−{1}\right\rbrace}.$$ Since (0, 0) is not in H. Thus H is not a subspace of $$R^{2}$$.
(b)Here it is given $$\displaystyle{H}={\left\lbrace{a}{t}+{b}:{b}={8}{a}\right\rbrace}.$$ Clearly (0, 0) is in H. Let $$at+b,ct+d \in H$$. Then $$b=8a\ and\ d=8c$$. Now for a scaler k