Linear Spaces, Mappings and Dimensional Spaces

1) Show that if dim X = 1 and T belongs to L(X,X), there exists k in K st Tx=kx for all x in X.

2) Let U and V be finite dimensional linear spaces and S belong to L(V,W), T belong to L(U,V).  Show that the dimension of the null space of ST is less than or equal to the sum of the dimensions of the null spaces of S and T.

3) Show that is X is a finite dimensional space then the space L(X,X) of all linear maps of X into X is finite dimensional.  Find the dimension of L(X,X).