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.