Give an example which satisfies the properties of vector space, subspace and inner product space and number should be complex
1
Expert's answer
2021-07-15T10:19:17-0400
Vector space:
A vector space is a set of objects which is called vectors, which may be added together and multiplied together by a numbers, called scalers.
Example:
V=c1u1+c2u2+c3u3
Subspace:
A subspace of a vector space V is a subset of H of V which have 3 properties -
the zero vector of v is in H
H is closed under the vector addition that is for each u and v in H, the sum u+v in H.
H is closed under multiplication by scalers that is for each u in H and each scaler c, the vector cu is in H.
Example:
u+v=v+u
(u+v)+w=u+(v+w)
u+(-u)=0
vector space V • {0}
The trivial space {0} is a subspace of V.
Ex. V=R2 .
The line x − y = 0 is a subspace of R2 .
Vector inner product space :
Let u, v and w be a vector in a vector space V and let c be any scaler. An inner product on V is function that associates a real number <u,v> with each pair of vector u and v and satisfies the following axioms -
Comments