Question

determine whether or not the set of vectors {(1,2,-1),(0,3,1),(1,-5,3)} is a basis of R^3 also find dimension

Answer 1

We can set up a matrix and use Gaussian elimination .

$$\begin{pmatrix} 1 & 2 & -1 \\ 0 & 3 & 1 \\ 1 & -5 & 3 \\ \end{pmatrix}$$

$R_3=R_3-R_1$

$$\begin{pmatrix} 1 & 2 & -1 \\ 0 & 3 & 1 \\ 0 & -7 & 4 \\ \end{pmatrix}$$

$R_2=R_2/3$

$$\begin{pmatrix} 1 & 2 & -1 \\ 0 & 1 & 1/3 \\ 0 & -7 & 4 \\ \end{pmatrix}$$

$R_1=R_1-2R_2$

$$\begin{pmatrix} 1 & 0 & -5/3 \\ 0 & 1 & 1/3 \\ 0 & -7 & 4 \\ \end{pmatrix}$$

$R_3=R_3+7R_2$

$$\begin{pmatrix} 1 & 0 & -5/3 \\ 0 & 1 & 1/3 \\ 0 & 0 & 19/3\\ \end{pmatrix}$$

$R_3=(3/19)R_3$

$$\begin{pmatrix} 1 & 0 & -5/3 \\ 0 & 1 & 1/3 \\ 0 & 0 & 1\\ \end{pmatrix}$$

$R_1=R_1+(5/3)R_3$

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

$R_2=R_2-R_3/3$

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

Since the rank of the matrix is 3, then the set {(1,2,-1),(0,3,1),(1,-5,3)} is a basis of $R^3.$

The dimension is 3.