## Linear Algebra and Its Applications, Review Exercise 2.1

Review exercise 2.1. For each of the following subspaces of $\mathbb{R}^4$ find a suitable set of basis vectors:

a) all vectors for which $x_1 = 2x_4$

b) all vectors for which $x_1+x_2+x_3 = 0$ and $x_3+x_4=0$

c) all vectors consisting of linear combinations of the vectors $(1, 1, 1, 1)$, $(1, 2, 3, 4)$, and $(2, 3, 4, 5)$

Answer: a) Since there are no constraints on $x_2$, $x_3$, and $x_4$ they may assume any value. (In other words, $x_2$, $x_3$, and $x_4$ are all free variables and only $x_1$ is a basic variable.) If $x_2 = x_3 = 0$ and $x_4 = 1$ then the  vector $(2, 0, 0, 1)$ is in the subspace and can serve as a basis vector. If $x_2 = 1$ and $x_3 = x_4 = 0$ then we obtain a second basis vector $(0, 1, 0, 0)$ and if $x_2 = x_4 = 0$ and $x_3 = 1$ then we obtain a third basis vector $(0, 0, 1, 0)$.

The vector $(2, 0, 0, 1)$, $(0, 1, 0, 0)$ and $(0, 0, 1, 0)$ thus form a basis for the subspace.

b) The two equations form the linear system $\begin{bmatrix} 1&1&1&0 \\ 0&0&1&1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = 0$

In this system $x_1$ and $x_3$ are basic variables and $x_2$ and $x_4$ are free variables. If we set $x_2 = 1$ and $x_4 = 0$ then we have $x_3 = 0$ and $x_1 = -1$. If we set $x_2 = 0$ and $x_4 = 1$ then we have $x_3 = -1$ and $x_1 = 1$.

The two vectors $(-1, 1, 0, 0)$ and $(1, 0, -1, 1)$ are thus basis vectors for the subspace (which happens to be the nullspace of the matrix above).

c) We have $(2, 3, 4, 5) = (1, 1, 1, 1) + (1, 2, 3, 4)$. So of the three vectors only two are linearly independent, and $(1, 1, 1, 1)$ and $(1, 2, 3, 4)$ 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 .

This entry was posted in linear algebra. Bookmark the permalink.