Mathematics II - Old Questions

17. Let U and V be vector spaces over a field and assume that dim U=dim V. If   is a linear transformation, then prove that the following are equivalent;

i. T is invertable

ii. T is one-one and onto, and 

iii. T is non-singular 

OR 

Verify that the set of matrices of the form  is a subspace of the vector space of 3 x 3 matrices.