Inner Product vs. Dot Product
Could someone briefly go over what the difference is between inner products and dot products?
jacobtsai0: June 2, 2015, 4:27 p.m.
jacobtsai0: June 2, 2015, 4:29 p.m.
The dot product is a particular example of an inner product. Inner product is a more general term than dot product, and can refer to other maps in other contexts.
Jackhe: June 2, 2015, 4:32 p.m.
So dot product is inner product but only in R^n?
alanvce: June 2, 2015, 4:40 p.m.
Yeah the dot product is when you are in standard coordinates. The inner product in general can be defined even on infinite dimensional vector spaces. The integral example is a good example of that.
alanvce: June 2, 2015, 4:41 p.m.
A dot product is a very specific inner product that works on R^n
TLindberg: June 2, 2015, 4:53 p.m.
I also believe the dot product needs to act in the standard basis correct?
JeffreyLam1: June 2, 2015, 5:29 p.m.
Think of the inner product like a rule that turns 2 elements in a space into a scalar. The inner product has to follow a bunch of rules, but most of them have useful purposes such as the dot product when we try to find a point on a plane closest to a vector. The book talks about the integral inner product that Weisbart has in his practice test too, and explains it as useful because it allows us to fit a function closest to the one in question.
alexwang0518: June 4, 2015, 1:44 p.m.
Some key things to note about inner product. 1) The dot product is the "standard inner product" i.e. you add the multiples together. 2) When you are given a differently defined inner product, this new inner product defines length and perpendicularity. So, when doing the GS Process, the dot product and the magnitude are swapped out. Remember the magnitude is the vector with itself. So if the inner product is the integral from A to B of fg dx, the magnitude of v would be the integral from A to B of vv dx.
Jerry_Zhongyang_Liu: June 5, 2015, 8:50 p.m.
Any inner product with an orthonormal basis can be treated as a dot product.