Answer:
a) Assume that
, and
is a scalar (a real or complex number).
<em>First. </em>Let us prove that
is not empty. This is easy because
, by linearity. Here,
stands for the zero vector of V, and
stands for the zero vector of W.
<em>Second.</em> Let us prove that
. By linearity
.
Then,
.
<em>Third. </em> Let us prove that
. Again, by linearity
.
And the statement readily follows.
b) Assume that
and
are in range of
. Then, there exist
such that
and
.
<em>First.</em> Let us prove that range of
is not empty. This is easy because
, by linearity.
<em>Second.</em> Let us prove that
is on the range of
.
.
Then, there exist an element
such that
. Thus
is in the range of
.
<em>Third.</em> Let us prove that
is in the range of
.
.
Then, there exist an element
such that
. Thus
is in the range of
.
Notice that in this second part of the problem we used the linearity in the reverse order, compared with the first part of the exercise.
Step-by-step explanation: