Linear Algebra with Applications, 10th Edition

Section 3.6 Exercises

1. For each of the following matrices, find a basis for the row space, a basis for the column space, and a basis for the null space:

   1. $\left[\begin{array}{ccc}1& 3& 2\\ 2& 1& 4\\ 4& 7& 8\end{array}\right]$

   2. $\left[\begin{array}{cccc}\hfill -3& 1& \hfill 3& \hfill 4\\ \hfill 1& 2& \hfill -1& \hfill -2\\ \hfill -3& 8& \hfill 4& \hfill 2\end{array}\right]$

   3. $\left[\begin{array}{cccc}\hfill 1& 3& \hfill -2& \hfill 1\\ \hfill 2& 1& \hfill 3& \hfill 2\\ \hfill 3& 4& \hfill 5& \hfill 6\end{array}\right]$

2. In each of the following, determine the dimension of the subspace of ${ℝ}^3$ spanned by the given vectors:

   1. $\left[\begin{array}{r}1\\ -2\\ 2\end{array}\right],\left[\begin{array}{r}2\\ -2\\ 4\end{array}\right],\left[\begin{array}{r}-3\\ 3\\ 6\end{array}\right]$

   2. $\left[\begin{array}{r}1\\ 1\\ 1\end{array}\right],\left[\begin{array}{r}1\\ 2\\ 3\end{array}\right],\left[\begin{array}{r}2\\ 3\\ 1\end{array}\right]$

   3. $\left[\begin{array}{r}1\\ -1\\ 2\end{array}\right],\left[\begin{array}{r}-2\\ 2\\ -4\end{array}\right],\left[\begin{array}{r}3\\ -2\\ 5\end{array}\right],\left[\begin{array}{r}2\\ -1\\ 3\end{array}\right]$

3. Let

   $$A=\left[\begin{array}{cccccc}1& 2& 2& 3& 1& 4\\ 2& 4& 5& 5& 4& 9\\ 3& 6& 7& 8& 5& 9\end{array}\right]$$

   1. Compute the reduced row echelon form U of A. Which column vectors of U correspond to the free variables? Write each of these vectors as a linear combination of the column vectors corresponding to the lead variables.

   2. Which column vectors of A correspond to the lead variables of U? These column vectors form a basis for the column space of A. Write ...