Bases for Vector Space V=\mathbb{R}^3

[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).