show that if u dot v = u dot w for all u, then v = w.
You can use the result of the previous problem:
http://www.jiskha.com/display.cgi?id=1184622247
Linearity of the inner product implies
u dot v = u dot w ----->
u dot(v-w) = 0
This is true for all u, therefore the result of the previous problem says that:
v - w = 0 ---->
v = w
To show that if u dot v = u dot w for all u, then v = w, we can use the result of the previous problem, which states that if u dot x = 0 for all u, then x = 0.
First, we start with the assumption that u dot v = u dot w for all u.
Using the linearity of the inner product, we can rewrite this as:
u dot v - u dot w = u dot (v - w)
Now, we want to show that u dot (v - w) = 0 for all u.
If we can show that u dot (v - w) = 0 for an arbitrary u, then we can invoke the result of the previous problem, which says that if u dot x = 0 for all u, then x = 0.
Therefore, for the given assumption, we have:
u dot (v - w) = 0, for all u.
Since this is true for all u, we can say that v - w = 0.
Therefore, v = w.
Hence, if u dot v = u dot w for all u, then v = w.