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For questions 1 through 5, let $$A = \left[\begin{array}{ccccc} 1 & 2 & 1 & 1 & -2\\ 2 & 4 & 1 & 0 & -1 \\ 0 & 0 & 0 & 1 & -2 \\ 0 & 0 & 1 & -2 & 5\end{array}\right]$$

1.  Find a basis for Nul(\(A\)).

a.  $$\left\{ \left[ \begin{array}{c} -2\\1\\0\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\0\\-1\\2\\1 \end{array}\right]\right\}$$

b.  $$\left\{ \left[ \begin{array}{c} -2\\1\\0\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\0\\1\\-2\\-1 \end{array}\right]\right\}$$

c.  $$\left\{ \left[ \begin{array}{c} -2\\1\\0\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\0\\1\\-2\\-1 \end{array}\right], \left[ \begin{array}{c} 0\\0\\1\\0\\-1 \end{array}\right]\right\}$$

d.  $$\left\{ \left[ \begin{array}{c} -2\\1\\0\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\0\\1\\-2\\-1 \end{array}\right], \left[ \begin{array}{c} 0\\3\\1\\0\\-2 \end{array}\right]\right\}$$

2.  Find a basis for Col(\(A\))

a.  $$\left\{ \left[ \begin{array}{c} 1\\0\\0\\0 \end{array}\right], \left[ \begin{array}{c} 0\\1\\0\\0 \end{array}\right], \left[ \begin{array}{c} 0\\0\\1\\0 \end{array}\right]\right\}$$

b.  $$\left\{ \left[ \begin{array}{c} 2\\0\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\-1\\2\\0 \end{array}\right]\right\}$$

c.  $$\left\{ \left[ \begin{array}{c} 2\\4\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\0\\1\\-2 \end{array}\right]\right\}$$

d.  $$\left\{ \left[ \begin{array}{c} 1\\2\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\1\\0\\1 \end{array}\right], \left[ \begin{array}{c} 1\\0\\1\\-2 \end{array}\right]\right\}$$

e.  $$\left\{ \left[ \begin{array}{c} 1\\2\\0\\0 \end{array}\right], \left[ \begin{array}{c} 2\\4\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\1\\0\\1 \end{array}\right], \left[ \begin{array}{c} -2\\-1\\-2\\5 \end{array}\right]\right\}$$

3.  Find a basis for Row(\(A\)). (Take the non-zero rows of your reduced-echelon form matrix)

a.  $$\left\{ \left[ \begin{array}{c} 1\\2\\1\\1\\-2 \end{array}\right], \left[ \begin{array}{c} 2\\4\\1\\0\\-1 \end{array}\right]\right\}$$

b.  $$\left\{ \left[ \begin{array}{c} 1\\2\\0\\0\\-1 \end{array}\right], \left[ \begin{array}{c} 0\\0\\1\\0\\1 \end{array}\right], \left[ \begin{array}{c} 0\\0\\0\\1\\-2 \end{array}\right]\right\}$$

c.  $$\left\{ \left[ \begin{array}{c} 1\\2\\0\\0\\-1 \end{array}\right], \left[ \begin{array}{c} 0\\0\\1\\0\\1 \end{array}\right], \left[ \begin{array}{c} 0\\0\\0\\1\\2 \end{array}\right]\right\}$$

4.  What is Rank(\(A\))? (This is just the number of vectors in a basis for the column space.)

a.  2

b.  3

c.  4

5.  Is \(A\) invertible?

a.  Yes.

b.  Go jump in the lake.