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.