Proving subset is a subspace of vector space R2

Question

$$W = \begin{bmatrix}a\\b\end{bmatrix} \in \mathbb{R}^2 : 2a − 3b = 0$$

I know the zero vector exists, but I need some help proving the addition and multiplication.

Thanks

Answer 1

1) As you say, the zero vector $\underline 0 \in W$ because $a=b=0 \implies 2a - 3b = 0$.

2) Suppose $\underline u = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} \in W$ and $\underline v = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}\in W$. Then $2 u_1 - 3u_2 = 0 = 2 v_1 - 3 v_2$. From this it follows that $2u_1 - 3u_2 + 2v_1 - 3v_2 = 2(u_1 + v_1) -3(u_2 + v_2) = 0$, so $\underline u + \underline v \in W$.

3) Now suppose that we have $\underline u$ as above, and $\alpha \in \mathbb{R}$. Then $\alpha \underline u$ = $\begin{bmatrix}\alpha u_1 \\ \alpha u_2 \end{bmatrix}$. Since $2u_1 - 3u_2 = 0$, we have $\alpha(2u_1 - 3u_2) = 2(\alpha u_1) - 3(\alpha u_2) = 0$ and so $\alpha \underline u \in W$.

So $W$ is a subspace of $\mathbb{R}^2$.

Answer 2

Hint

$$2(a_1+a_2)-3(b_1+b_2)= (2a_1-3b_1) + ( 2a_2-3b_2)$$ Also $$ 2\lambda a -3\lambda b =\lambda (2a-3b)$$