Question #215181

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+c3u3V=c_1u_1+c_2u_2+c_3u_3

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=R2V = R_2 .

The line x − y = 0 is a subspace of R2R_2 .

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 -

<u,v>=<v,u>

<u,v+w>=<u,v>+<u,w>

c<u,v>=<cu,v>

<u,u> and <v,v>=0 if v=0

Example:

A,B=tr(BTA)〈A, B〉= tr(B^TA)

A=[a11a12a21a22a31a32]A=\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \\ \end{bmatrix}


B=[b11b12b21b22b31b32]B=\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ b_{31} & b_{32} \\ \end{bmatrix}


BTA=A=[a11b11+b21a21+b31a31b11a12+b21a22+b31a32b12a11+b22a21+b32a31b12a12+b22a22+b32a32]B^TA=A=\begin{bmatrix} a_{11} b_{11}+b_{21}a_{21}+b_{31}a_{31}& b_{11} a_{12}+b_{21}a_{22}+b_{31}a_{32} \\ b_{12} a_{11}+b_{22}a_{21}+b_{32}a_{31}& b_{12} a_{12}+b_{22}a_{22}+b_{32}a_{32} \\ \end{bmatrix}


<A,B>=b11a11+b21a21+b31a31+b12a12+b22a22+b32a32<A,B>=b_{11}a_{11}+b_{21}a_{21}+b_{31}a_{31}+b_{12}a_{12}+b_{22}a_{22}+b_{32}a_{32}


=Σi=13Σj=13aijbij=\Sigma_{i=1}^{3}\Sigma_{j=1}^{3}a_{ij}b_{ij}

Hence, we can say that the inner product space (M3,2,<,>)(M_{3,2,<,>}) is isomorphic to the euclidean space (R3×2,.)(R^{3\times 2},.)


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