###### Activity3.2.2
1. In the preview activity, we considered a set of vectors in $$\real^2\text{:}$$

\begin{equation*} \vvec_1 = \twovec{2}{1}, \vvec_2 = \twovec{1}{2}\text{.} \end{equation*}

Explain why these vectors form a basis for $$\real^2\text{.}$$

2. Consider the set of vectors in $$\real^3$$

\begin{equation*} \vvec_1 = \threevec{1}{1}{1}, \vvec_2 = \threevec{0}{1}{-1}, \vvec_3 = \threevec{1}{0}{-1}\text{.} \end{equation*}

and determine whether they form a basis for $$\real^3\text{.}$$

3. Do the vectors

\begin{equation*} \vvec_1 = \threevec{-2}{1}{3}, \vvec_2 = \threevec{3}{0}{-1}, \vvec_3 = \threevec{1}{1}{0}, \vvec_4 = \threevec{0}{3}{-2} \end{equation*}

form a basis for $$\real^3\text{?}$$

4. Explain why the vectors $$\evec_1,\evec_2,\evec_3$$ form a basis for $$\real^3\text{.}$$

5. If a set of vectors $$\vvec_1,\vvec_2,\ldots,\vvec_n$$ forms a basis for $$\real^m\text{,}$$ what can you guarantee about the pivot positions of the matrix

\begin{equation*} \left[\begin{array}{rrrr} \vvec_1 \amp \vvec_2 \amp \ldots \amp \vvec_n \end{array}\right]\text{?} \end{equation*}
6. If the set of vectors $$\vvec_1,\vvec_2,\ldots,\vvec_n$$ is a basis for $$\real^{10}\text{,}$$ how many vectors must be in the set?

in-context