Problem 1
Let
be a linear operator on a finite-dimensional vector space
. Let
be a basis of
and
.
Suppose that
is an invertible
matrix. Show that there
is a basis
of
such that
.
Problem 2
Let
be a linear transformation.
a) (
Rank-Nullity Theorem) If
is finite-dimensional, prove that
; (Give a direct
proof, without using theorem of section 49 in the book)
b) Give an example of an infinite-dimensional vector space and a linear
operator on such that the statement of the rank-nullity theorem
is false. Return to your proof of this theorem and think why it does not work
in the infinite-dimensional case.
Problem 3
Prove that a linear transformation is one-to-one iff its null space consists
of the zero vector only.
Problem 4
Prove that a linear transformation
, where
and
are finite-dimensional vector spaces with
,
is one-to-one if and only if
. (
Hint:
first, prove that it is one-to-one iff it is onto; then prove that it is onto
iff
).