Question

let v
be the vector
space of polynomials with
linear product given by <f,g>=
where f(t)=t+2 and g(t)=t²-2t-3. Find (f,g).​

Answer 1

Answer:

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Answer 2

Answer:

Parallelogram law, the ability to measure angle between two vectors and in particular, the

concept of perpendicularity make the euclidean space quite a special type of a vector space.

Essentially all these are consequences of the dot product. Thus, it makes sense to look for

operations which share the basic properties of the dot product. In this section we shall

briefly discuss this.

Definition 6.1 Let V be a vector space. By an inner product on V we mean a binary

operation, which associates a scalar say hu, vi for each pair of vectors u, v) in V, satisfying

the following properties for all u, v, w in V and α, β any scalar. (Let “−” denote the complex

conjugate of a complex number.)

(1) hu, vi = hv, ui (Hermitian property or conjugate symmetry);

(2) hu, αv + βwi = αhu, vi + βhu, wi (sesquilinearity);

(3) hv, vi > 0 if v 6= 0 (positivity).

A vector space with an inner product is called an inner product space.