Exercise 4

Consider the following vectors in $\real^3$: $$\vvec_1=\threevec{1}{3}{2}, \vvec_2=\threevec{0}{1}{4}, \vvec_3=\threevec{-2}{-5}{0}, \vvec_4=\threevec{-2}{-1}{-1}, \vvec_5=\threevec{1}{-2}{-1}$$.

1. Do these vectors form a basis for $\real^3$? Explain your thinking.

2. Find a subset of these vectors that forms a basis of $\real^3$.

3. Suppose you have a set of vectors $\vvec_1, \vvec_2,\ldots,\vvec_6$ in $\real^4$ such that $$\left[\begin{array}{rrrr} \vvec_1 & \vvec_2 & \ldots & \vvec_6 \end{array}\right] \sim \left[\begin{array}{rrrrrr} 1 & 0 & -2 & 0 & 1 & 0 \\ 0 & 1 & 3 & 0 & -4 & 0 \\ 0 & 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array}\right]$$. Find a subset of the vectors that forms a basis for $\real^4$.