Page 207 - A Course in Linear Algebra with Applications

6.3: Kernel, Image and Isomorphism 191

2. Show that every subspace U of a finite-dimensional vector
space V is the kernel and the image of suitable linear operators
on V. [Hint: assume that U is non-zero, choose a basis for U
and extend it to a basis of V].
3. Sort the following vector spaces into batches, so that those
within the same batch are isomorphic:
6
6
R , R 6 , C , P 6 (C),M 2 , 3 (R),C[0,1].
4. Show that a linear transformation T : V —> W is injective if
and only if it has the property of mapping linearly independent
subsets of V to linearly independent subsets of W.
5. Show that a linear transformation T : V —• W is surjec-
tive if and only if it has the property of mapping any set of
generators of V to a set of generators of W.
6. A linear operator on a finite-dimensional vector space is
an isomorphism if and only if some representing matrix is
invertible: prove or disprove.
7. Prove that the composite of two linear transformations is
a linear transformation.
8. Prove parts (i) and (ii) of Theorem 6.3.10.
9. Let T : V —> W and S : W —> U be isomorphisms of
vector spaces; show that the function ST : V —> U is also an
isomorphism.
10. Let T be a linear operator on a finite-dimensional vector
space V. Prove that the following statements about T are
equivalent:
(a) T is injective;
(b) T is surjective;
(c) T is an isomorphism.
Are these statements still equivalent if V is infinitely gener-
ated?
11. Show that similar matrices have the same rank. [Use the
fact that similar matrices represent the same linear operator].