Inner products?

Question

I'm studying linear algebra, and i've decided that inner products are the strangest things i've seen in a very long time. I can do the math, but I don't understand what applications they serve. can anyone tell me?

Answer 1

Inner products allow you to calculate the length of a vector and the angle between two vectors--very important stuff for linear algebra.

Answer 2

When or how did you arrive at that conclusion?

How can you say that inner product spaces are the strangest things you have ever seen? Inner product spaces are just vector spaces with an inner product defined on them. Vectors are nice, very intuitive. You, my friend, need to take some calculus-based Physics classes. Trust me, I went from a C to an A just because I took some Physics. Physics really helps you understand all of Calculus, Linear Algebra, and Differential Equations. Physics helps answer all of your questions that you asked.

Vectors have two components, magnitude and direction. Inner products help us with both of these concepts. We can find the magnitude of a vector and inner products tell us about orthogonality, if two vectors are perpindicular to each other. Inner products are like our rulers which let us measure our vectors.

Euclidean spaces (and even then R1, R2, and R3) are the only ones which make sense in the real physical world. The only inner product which is useful is the standard dot product.

But in math, there are many other inner product spaces defined so they just teach about such things even though it may not be obvious where in the real world this maybe used.

For mathematicians, R9 and the set of all 3x3 matrices are exactly the same. A 3x3 matrix is a vector. How in the world are you going to measure the length of the 3x3 matrix? How can you measure the angle between two matrices? If you think in terms of the real world, you would just boggle your mind.