Show $\langle u,v \rangle = u_1v_1+3u_2v_2$ is an inner product

Question

I understand for a vector space to be an inner product it must meet four properties:

1. $\langle u+v,w \rangle =\langle u,w \rangle + \langle v,w \rangle$.

2. $\langle \alpha v,w \rangle = \alpha \langle v,w \rangle$

3. $\langle u , v \rangle = \langle v, u \rangle $.

4. $\langle v,v \rangle \geq 0 $ with equality iff $v=0$.

   But I'm not sure how I can go about proving the above equation as an inner product.

Answer 1

1. $\langle u+v,w \rangle = (u_1+v_1)w_1+3(u_2+v_2)w_2 = (u_1 w_1 +3u_2 w_2)+ (v_1 w_1 +3v_2 w_2)\\ =\langle u,w \rangle + \langle v,w \rangle$

2. $ \langle \alpha v,w \rangle = (\alpha v_1) w_1 +3(\alpha v_2) w_2 = \alpha (v_1 w_1 +3v_2 w_2) =\alpha \langle v,w \rangle$

The rest is that easy, just expand and distribute ;)