Which are bases?

[Ted123]

Homework Statement

Which of the following sets $S$ are bases for the vector space $V=\mathbb{R}^3$?

(a) $S=\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \right\}$

(b) $S=\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \right\}$

(c) $S=\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} , \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix} \right\}$

(d) $S=\left\{ \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} , \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix} \right\}$

The Attempt at a Solution

By my reckoning, the only set that isn't a basis is (b) as it isn't linearly independent and $\text{Span}(S)\neq V$. Correct?

 

[LCKurtz]

Since there are only 3 and the dimension of R³ is 3, you are correct that whether or not they are linearly independent determines whether they are a spanning set (basis).