I dont think they are asking for the dot product but to define that these formulas satsify the axioms of an inner product. $\endgroup$ – Q.matin Commented Sep 23, 2012 at 7:04

You want to verify all the properties of a real inner product (since we're looking at a real vector space). Using your notation $\langle A,B \rangle = \mathrm{tr}(B^T A)$, we want to check that: …

Inner products are related to the study of "angles". The "dot" product of 3D vectors is an inner product, and it has many uses. An inner product is often used to define a "norm" function that …

The definition of an inner product on the vector space of let's say continuous functions on $[0,1]$ of $$\langle f, g \rangle = \int_0^1 f(x)g(x) \ dx$$ works in that it satisfies those axioms. Now, …

Dec 28, 2017 · The moral of the story is that both $$ (x,y) \mapsto x^Ty \qquad\text{and}\qquad (x,y) \mapsto x^T A y $$ can define inner products on $\mathbb{R}^n$. Indeed, Sean …

Inner product $\langle u,v\rangle$ doesn't depend on the choice of basis. In the given basis $\langle u,v \rangle =x^TMy$.

Then, prove that the inner product is additive in the first component: $\langle x+u,v\rangle = \langle x,v\rangle + \langle u,v\rangle$. Then, prove the result holds for $\lambda$ any positive …

For example, the concept of orthogonality comes from inner product being zero, but in a topological space, one cannot always define an inner product (since an inner product space is …

Oct 24, 2014 · What does inner product actually mean? So far most of the cases that I encounter seems to suggest that dot product is the only useful inner product. I mean most of the things …

You mean: Define a new / second inner product such that two vectors orthogonal with respect to the first / old inner product are not orthogonal with respect to the new / second one? Yes! That …