# Showing that the set of points is a subspace

I given two vectors here:

$V = \mathbb R^3$

$W = \{(a+b,b+c,a-c)|a,b,c\in \mathbb R^3 \}$

Again, I am trying to prove or disprove if $W$ is a subspace of $V$.

For the $0$ vector, I chose $a=0$,$b=0$ and $c=0$ as my $0$ vector is in W so the first condition is satisfied.

For addition, I picked the points $R$=$(a_1+b_1,b_1+c_1,a_1-c_1)$ and $S$=$(a_2+b_2,b_2+c_2,a_2-c_2)$. If I add them, I get

$T$=$(\{a_1+a_2\}+\{b_1+b_2\},\{b_1+b_2\}+\{c_1+c_2\},\{a_1+a_2\}-\{c_1+c_2\})$

which I can just write as single element

$T$=$(\{a_3\}+\{b_3\},\{b_3\}+\{c_3\},\{c_3\}-\{c_3\})$

which I believe is closed under scalar addition since each combination of those points can be written as a single element.

Finally for scalar multiplication, if I take a scalar $c$ and multiply it by all 3 points, that will give me a new element in the set of points which is also contained in $W$ and thus, W is a subspace of $V$.

Is that the correct approach? Sorry for the novice questions. I'm trying to do linear algebra after 4 years and it has not been the best.

• Yes, I think it will qualify as a subspace in that case Commented Mar 5, 2017 at 9:56
• You are welcome. Commented Mar 5, 2017 at 10:08

Yes, $W$ is the image space of the linear map with matrix
$$\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & -1 \end{pmatrix}$$
from $\mathbb{R}^3$ to itself. General fact: image spaces of linear maps are subspaces.