## Determine Whether Each Set is a Basis for $\R^3$

## Problem 579

Determine whether each of the following sets is a basis for $\R^3$.

**(a)** $S=\left\{\, \begin{bmatrix}

1 \\

0 \\

-1

\end{bmatrix}, \begin{bmatrix}

2 \\

1 \\

-1

\end{bmatrix}, \begin{bmatrix}

-2 \\

1 \\

4

\end{bmatrix} \,\right\}$

**(b)** $S=\left\{\, \begin{bmatrix}

1 \\

4 \\

7

\end{bmatrix}, \begin{bmatrix}

2 \\

5 \\

8

\end{bmatrix}, \begin{bmatrix}

3 \\

6 \\

9

\end{bmatrix} \,\right\}$

**(c)** $S=\left\{\, \begin{bmatrix}

1 \\

1 \\

2

\end{bmatrix}, \begin{bmatrix}

0 \\

1 \\

7

\end{bmatrix} \,\right\}$

**(d)** $S=\left\{\, \begin{bmatrix}

1 \\

2 \\

5

\end{bmatrix}, \begin{bmatrix}

7 \\

4 \\

0

\end{bmatrix}, \begin{bmatrix}

3 \\

8 \\

6

\end{bmatrix}, \begin{bmatrix}

-1 \\

9 \\

10

\end{bmatrix} \,\right\}$