I am having some issues with this problem in my Linear Algebra textbook. The goal is to either show that the given set, W, is a vector space, or to find a specific example to the contrary:
\begin{Bmatrix}
\begin{bmatrix}
a\\
b\\
c\\
d
\end{bmatrix} :
\begin{matrix}
3a + b = c\\
a + b + 2c = 2d
\end{matrix}
\end{Bmatrix}
I understand the basic properties of Vector Spaces – such as having to contain the zero vector, being closed under addition, and being closed under scalar multiplication. I have no problem proving when these sets are not vector spaces, for example if they do not contain the zero vector.
This set appears to contain the zero vector (if you plug in 0 for a, b, c, and d, the equations are consistent). But I'm not quite sure how to prove that this set is a vector space, or how to prove that it is closed under addition and scalar multiplication.
Thanks for your help.
Best Answer
If you have a vector satisfying the two constraints then multiplying by $k$ you get $$3a+b=c \implies 3(ka)+(kb)=(kc)$$ $$ a + b + 2c = 2d \implies (ka) + (kb) + 2(kc) = 2(kd)$$ so you have closure under scalar multiplication. Similarly for addition $$3a_1+b_1=c_1 \text{ and }3a_2+b_2=c_2 \\ \implies 3(a_1+a_2)+(b_1+b_2)=(c_1+c_2)$$ $$a_1 + b_1 + 2c_1 = 2d_1 \text{ and }a_2 + b_2 + 2c_2 = 2d_2 \\ \implies (a_1+a_2) + (b_1+b_2) + 2(c_1+c_2) = 2(d_1+d_2)$$