# Help proving a subset is a subspace

1. Apr 16, 2009

### paulrb

1. The problem statement, all variables and given/known data

Prove that the set of all 3-vectors orthogonal to [1, -1, 4] forms a subspace of R^3.

2. Relevant equations

Orthogonal means dot product is 0.

3. The attempt at a solution

I know the vectors in this subspace are of the form
[a,b,c] where a - b + 4c = 0.
However I don't know how to use this to show there is closure under vector addition and scalar multiplication.

2. Apr 16, 2009

### dx

Let v_1 and v_2 be two vectors orthogonal to the given vector (call it a), i.e.

$$v_1 \cdot a = 0$$
$$v_2 \cdot a = 0$$

Now, using these, all you have to do is show that

$$(v_1 + v_2) \cdot a = 0$$

HINT: Use distributivity of the dot product.