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).
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if dim X = 1, then X is spanned by some nonzero vector, which is clearly linearly independent, ...
Linear Spaces, Mappings and Dimensional Spaces are investigated.