Answer to Question #206221 in Linear Algebra for Aiman Arif

Question

show that the set {(1,2,3),(0,1,2),(0,0,1) of vectors generates or span R^3

Answer 1

We can set up a matrix and use Gaussian elimination to figure out the dimension of the space they span.

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

$R_1=R_1-2R_2$

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

$R_1=R_1+R_3$

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

$R_2=R_2-2R_3$

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

Since the rank of the matrix is 3, then the set {(1,2,3),(0,1,2),(0,0,1) of vectors generates or span $R^3.$