Activity 3.2.2

We will look at some examples of bases in this activity.

1. In the preview activity, we worked with the set of vectors in $\real^2$: $$\vvec_1 = \twovec{2}{1}, \vvec_2 = \twovec{1}{2}$$. Explain why these vectors form a basis for $\real^2$.

2. Consider the set of vectors in $\real^3$ $$\vvec_1 = \threevec{1}{1}{1}, \vvec_2 = \threevec{0}{1}{-1}, \vvec_3 = \threevec{1}{0}{-1}$$ and determine whether they form a basis for $\real^3$.

3. Do the vectors $$\vvec_1 = \threevec{-2}{1}{3}, \vvec_2 = \threevec{3}{0}{-1}, \vvec_3 = \threevec{1}{1}{0}, \vvec_4 = \threevec{0}{3}{-2}$$ form a basis for $\real^3$?

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

5. If a set of vectors $\vvec_1,\vvec_2,\ldots,\vvec_n$ forms a basis for $\real^m$, what can you guarantee about the pivot positions of the matrix $$\left[\begin{array}{rrrr} \vvec_1 & \vvec_2 & \ldots & \vvec_n \end{array}\right]$$?

6. If the set of vectors $\vvec_1,\vvec_2,\ldots,\vvec_n$ is a basis for $\real^{10}$, how many vectors must be in the set?