I needed clarification on a linear algebra question that I had:
Given the matrices $v_1 = \begin{bmatrix}
1 \\
1 \\
1 \\
\end{bmatrix}, $ $v_2 = \begin{bmatrix}
1 \\
-1 \\
1 \\
\end{bmatrix}$ and $v_3 = \begin{bmatrix}
1 \\
1 \\
-1 \\
\end{bmatrix}$,
1) How many vectors does the set {${v_1, v_2, v_3}$} have?
2) How many vectors are in Span{$v_1, v_2, v_3$}?
I think the answer to #1 is 3, simply because there are three matrices, and the answer to #2 is infinite, since there are an infinite number of linear combinations that can be made using these vectors.
I am uncertain of these answers, though.
Best Answer
Construct the matrix $$ \mathbf{A} = \left[ \begin{array}{ccc} v_{1} & v_{2} & v_{3} \end{array}\right] = \left[ \begin{array}{rrr} \phantom{-}1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1 \end{array}\right] . $$ Because $\det \mathbf{A} = 4 \ne 0$, the vector set is linearly independent.
The span of the vector set is $$ \text{span} \left\{ v_{1}, v_{2}, v_{3} \right\} = \mathbb{R}^{3}. $$ These vectors are a linearly independent span, also called a minimal spanning set for $\mathbb{R}^{3}$.