inner product space definition

Question

I have some problem in the definition of inner product space. The book I use to learn in linear algebra and its application 4th edition (David C.Lay)

In the chapter 6.7 it define the inner product space is an vetor space with an inner product. Also, in its example the inner product of the vetor $u=(u_1,u_2)$ and $v=(v_1,v_2)$ is set that $$<u,v>=4u_1v_1 + 5u_2v_2$$ Obviously, this satisfy all the axioms of an inner product. However in it previous chapter (chap 6.1) it define an inner product of 2 vector u and v to be $$<u,v>=u_1v_1 + u_2v_2$$ That's the point I don't understand. For me the latter make sense while the former seems a bit confusing. Does the inner product of the two vector u and v is defined by its entries $(u_1,u_2)$ and $(v_1,v_2)$ but why the inner product of the former is $4u_1v_1 + 5u_2v_2$.

Moreover, the definition of an inner product space is a vector space with an inner product. Does this mean that $R^n$ with any real number (inner product) can be called an inner product space. For example: $R^n$ and 3 is an inner product space ?

Can anyone explain it more specific with some detailed example

Answer 1

The two functions you gave are both inner products (since they satisfy the inner product axioms). The second one is the usual one. The first one is given as an example (I suppose), to show that many different inner products are possible.

If you equip $R^n$ with any inner product, then it becomes an inner product space. If the specific inner product is not mentioned, you can assume that people are talking about the usual one (the second one you showed).

Answer 2

An inner product is simply a mapping, it takes two elements of the vector space and returns an element of the base field of the vector space. It must satisfy the axioms of an inner product. That is, given an vector space $V$ over a field $F$ (where $F$ is a subfield of $\mathbb{R}$, although the definition can be generalized to $\mathbb{C}$ with some modifications), an inner product is a mapping: $$ \langle \cdot, \cdot \rangle: V \times V \to F $$

Such that for any three vectors $u, v, w \in V$ and any scalar $t \in F$, the following hold:

• $\langle u, v\rangle = \langle v, u\rangle$ (Symmetric)
• $\langle tu + v , w \rangle$ = $t\langle u, w \rangle + \langle v , w \rangle$ (Is a linear transformation with the right argument fixed)
• $\langle u, tv + w \rangle$ = $t\langle u, v \rangle + \langle u , w \rangle$ (Is a linear transformation with the left argument fixed)
• $\langle u, u\rangle$ is a nonnegative real number
• $\langle u, u\rangle = 0 \iff u = \mathbf{0}$ (Only the $0$-vector vanishes under the inner product)

If you equip a vector space with any binary mapping that satisfies the above, then you have an inner product space. An element of a vector space is not a binary operator, so it cannot be an inner product.

The first example you mentioned, since it satisfies the axioms, is an inner product. There are some useful properties of having an inner product associated with your vector space. One of them is that you can define a notion of distance. Sometimes don't want to use the Euclidean norm; for some applications a weighted norm is more useful, like determining a students mark given a set of assignments. In this case, the first example or a variant thereof, could be used.

Also, it is possible to define a notion of angle in these inner-product spaces by defining two vectors $v,w$ to be perpendicular iff $\langle v,w\rangle = 0$. Also, one could define the angle $\theta$ between two vectors $u,v$ to be related in the canonical way (if the situation permits): $\cos \theta := \dfrac{\langle u, v\rangle}{|u|\cdot|v|}$