Answer to Question #215181 in Linear Algebra for Haseeb

Question

Give an example which satisfies the properties of vector space, subspace and inner product space and number should be complex

Answer 1

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:

$V=c_1u_1+c_2u_2+c_3u_3$

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. $V = R_2$ .

The line x − y = 0 is a subspace of $R_2$ .

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:

$〈A, B〉= tr(B^TA)$

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

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

$B^TA=A=\begin{bmatrix} a_{11} b_{11}+b_{21}a_{21}+b_{31}a_{31}& b_{11} a_{12}+b_{21}a_{22}+b_{31}a_{32} \\ b_{12} a_{11}+b_{22}a_{21}+b_{32}a_{31}& b_{12} a_{12}+b_{22}a_{22}+b_{32}a_{32} \\ \end{bmatrix}$

$<A,B>=b_{11}a_{11}+b_{21}a_{21}+b_{31}a_{31}+b_{12}a_{12}+b_{22}a_{22}+b_{32}a_{32}$

$=\Sigma_{i=1}^{3}\Sigma_{j=1}^{3}a_{ij}b_{ij}$

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