Number of vectors in a set & span of a set 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.
Construct the matrix 
$$
\mathbf{A} = \left[ \begin{array}{ccc}
  v_{1} &amp; v_{2} &amp; v_{3} 
\end{array}\right]
 = 
\left[ \begin{array}{rrr}
    \phantom{-}1 &amp;  1 &amp;  1 \\ 
    1 &amp; -1 &amp;  1 \\ 
    1 &amp;  1 &amp; -1 
\end{array}\right] .
$$
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 $\mathbb{R}^{3}$.
Construct the matrix 
$$
\mathbf{A} = \left[ \begin{array}{ccc}
  v_{1} &amp; v_{2} &amp; v_{3} 
\end{array}\right]
 = 
\left[ \begin{array}{rrr}
    \phantom{-}1 &amp;  1 &amp;  1 \\ 
    1 &amp; -1 &amp;  1 \\ 
    1 &amp;  1 &amp; -1 
\end{array}\right] .
$$
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 $\mathbb{R}^{3}$.