Linear Algebra subspace questions

Question

These are my two questions! I don't even know how to set up these two questions!

For each of the following problems, is $\mathcal{W}$ a subspace of the given vector space $V$ over the given scalar field $\mathbb{F}$? Explain your reasoning!

$$(a)\quad V = \mathbb{P, \ F = R},\ \mathcal{W}=\{ax^3 + bx^2 + cx + d:\quad c + d = a + b\}$$

$$\text{($\mathbb{P}$ is the vector space for all polynomials over $\mathbb{R}$)}.$$

$$(b)\quad V = \mathbb{R^3, F =R}, \mathcal{W} = \left\{\begin{bmatrix} x\\ y\\ z\\\end{bmatrix} : 3x + 5y + 4z - 3 = 0\right\} $$

Answer 1

In general, given a vector space $V$ and a subset $W$, $W$ is a subspace of $V$ provided 1) $W$ is not empty, and 2) for all $p$ and $q$ in $W$, and all real numbers $\alpha$ and $\beta$, $\alpha p+\beta q$ is in $W$.

So, in (a), first show $W$ is not empty, then let $p$ and $q$ be typical elements of $W$, let $\alpha$ and $\beta$ be real numbers, and see if you can prove that $\alpha p+\beta q$ is in $W$.

(b) is a bit easier --- there is one element that is guaranteed to be in every vector space, and you should know what that element is, and you should be able to tell whether it's in $W$.