Review exercise 2.1. For each of the following subspaces of 
find a suitable set of basis vectors:
a) all vectors for which 
b) all vectors for which 
and 
c) all vectors consisting of linear combinations of the vectors 
, 
, and 
Answer: a) Since there are no constraints on 
, 
, and 
they may assume any value. (In other words, , , and are all free variables and only 
is a basic variable.) If 
and 
then the vector 
is in the subspace and can serve as a basis vector. If 
and 
then we obtain a second basis vector 
and if 
and 
then we obtain a third basis vector 
.
The vector , and thus form a basis for the subspace.
b) The two equations form the linear system
In this system and are basic variables and and are free variables. If we set and 
then we have 
and 
. If we set 
and then we have 
and 
.
The two vectors 
and 
are thus basis vectors for the subspace (which happens to be the nullspace of the matrix above).
c) We have 
. So of the three vectors only two are linearly independent, and and can serve as a basis for the subspace spanned by the vectors.
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
.
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