Exercise 2.3.4. Suppose the vectors ,
, and
are linearly independent. (The book says
linearly dependent
, but I believe this is a typo.) Are the vectors ,
, and
also linearly independent?
Answer: Consider the linear combination of ,
, and
with weights
,
, and
. We have
Since the vectors ,
, and
are linearly independent the above expression can be zero only if
,
, and
.
We can express this as the following system of equations
and can solve it via elimination. We first subtract the first equation from the second to obtain the following system:
and then add the second equation to the third to obtain the final system:
Solving for the weights we have or
,
or
, and
or
.
Since only if
we see that
,
, and
are linearly independent.
NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.
If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition
and the accompanying free online course, and Dr Strang’s other books
.