Problem 1:
If
are linearly independent vectors in
, show that
is linearly independent set.
Proof 1:
If
where
,
since
is linearly independent.
Is the proof correct? I felt something missing in the proof.
Please check you haven't yet proved anything... . You could argue as follows: Suppose since we're given is lin. ind.
Problem 2:
If
are linearly dependent vectors in
, show that
is linearly dependent set for any
.
Proof 2:
If
. Since
are linearly dependent,
for some
. Thus,
is linearly dependent.
Please check my proofs.
This is similar as the above one but at least here you mentioned that the given set of vectors is l.i....I'd wrap it up as follows: Since is l.d. we have a trivial linear combination and for some and STILL for some index is l.d.
Problem 3:
Let
and
,
is invertible .
Show that
are linearly independent if and only if
are linearly independent vectors in
.
How to prove problem 3? what theorem do I need?