# Thread: [SOLVED] Positive Definite Inner Product proofs

1. ## [SOLVED] Positive Definite Inner Product proofs

Let V be a vector space and < , > be a positive definite inner product. Then show that:
a) < 0, v > = 0 for any v in V
b) Show that for a fixed u in V, < u, v > = 0 for any v in V, then u = 0.
c) Show that < v, w > = 0 for any w in V implies that v = 0.

I did part a) like this:
< 0, v > = < 0 + 0, v > = < 0, v > + < 0, v>
< 0, v > = < 0, v > + < 0, v >
< 0, v > - < 0, v > = < 0, v >
0 = < 0, v>

Now I have been trying to prove parts b) and c) but I cannot think of a way to start. The statements both seem so obvious but I don't know where to begin with a proof.

2. Originally Posted by Zalren
Let V be a vector space and < , > be a positive definite inner product. Then show that:
a) < 0, v > = 0 for any v in V
b) Show that for a fixed u in V, < u, v > = 0 for any v in V, then u = 0.
In particular, take v= u.

c) Show that < v, w > = 0 for any w in V implies that v = 0.
Same suggestion- take w= v.

I did part a) like this:
< 0, v > = < 0 + 0, v > = < 0, v > + < 0, v>
< 0, v > = < 0, v > + < 0, v >
< 0, v > - < 0, v > = < 0, v >
0 = < 0, v>

Now I have been trying to prove parts b) and c) but I cannot think of a way to start. The statements both seem so obvious but I don't know where to begin with a proof.

3. Thanks! I feel like it was too easy once you helped me with the first step. I was just stuck for some reason.