申論題

Notations: 

■Rmxn: the set of m x n matrices with entries in R.

■ : the transpose (inverse) of matrix A.

■ det(A): the determinant of matrix A.

■ Null(A): the null space of matrix A.

■ rank(A) (nullity(A)): the rank (nullity) of matrix A.

■ I_n: the identity matrix of size n x n.

■: the matrix representation of T relative to a and β.

 (1) True or False (If true, prove it; if false, give a counterexample or disprove it).

(vii) (5%) In R³, if we define the inner product <,> by
then R³ with this defined inner product is an inner-product space.