Question

Consider the set B={(1,1,1),(0,2,2),(0,0,3)}.show that B

I) spans R³

ii)is linearly independent

iii)is a basis for R³

Answer 1

i)

$$\begin{pmatrix} 1 & 0 & 0 \\ 1 & 2 & 0 \\ 1 & 2 & 3 \\ \end{pmatrix}$$

$R_2=R_2-R_1$

$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 1 & 2 & 3 \\ \end{pmatrix}$$

$R_3=R_3-R_1$

$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 2 & 3 \\ \end{pmatrix}$$

$R_2=R_2/2$

$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 2 & 3 \\ \end{pmatrix}$$

$R_3=R_3-2R_2$

$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \\ \end{pmatrix}$$

$R_3=R_3/3$

$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$$

The rank of the matrix is 3, so the given vectors span a subspace of dimension 3, hence they span R³.

ii)

$$a\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}+b\begin{pmatrix} 0 \\ 2 \\ 2 \end{pmatrix}+c\begin{pmatrix} 0 \\ 0 \\ 3 \end{pmatrix}=\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$$

$$\begin{pmatrix} 1 & 0 & 0 & & 0 \\ 1 & 2 & 0 & & 0 \\ 1 & 2 & 3 & & 0 \\ \end{pmatrix}\to\begin{pmatrix} 1 & 0 & 0 & & 0 \\ 0 & 1 & 0 & & 0 \\ 0 & 0 & 1 & & 0 \\ \end{pmatrix}$$

$a=b=c=0$

The given vectors are linearly independent.

iii) A subset $S$ of a vector space $V$  is called a basis if

1. $S$ is a spanning set

2. $S$ is linearly independent.

Therefore the set $B=\{{(1,1,1),(0,2,2),(0,0,3)\}}$ is a basis for R³.