Directions: You may work to solve these problems in groups, but all written work must be your own. Show all work; no credit for solutions without work..

1. [2.8.{15-17}] Determine which sets below are bases for ${ℝ}^2$ or ${ℝ}^3$. Justify each answer.

   1. $\left[\begin{array}{c}\hfill 5\\ \hfill -2\end{array}\right],\left[\begin{array}{c}\hfill 10\\ \hfill -3\end{array}\right]$
   2. $\left[\begin{array}{c}\hfill -4\\ \hfill 6\end{array}\right],\left[\begin{array}{c}\hfill 2\\ \hfill -3\end{array}\right]$
   3. $\left[\begin{array}{c}\hfill 0\\ \hfill 1\\ \hfill -2\end{array}\right],\left[\begin{array}{c}\hfill 5\\ \hfill -7\\ \hfill 4\end{array}\right],\left[\begin{array}{c}\hfill 6\\ \hfill 3\\ \hfill 5\end{array}\right]$

2. [2.8.{23-25}] Given a matrix $A$ and an echelon form of $A$, find a basis for $\mathrm{Col}(A)$ and $\mathrm{Nul}(A)$.

   1. $A=\left[\begin{array}{cccc}\hfill 4& \hfill 5& \hfill 9& \hfill -2\\ \hfill 6& \hfill 5& \hfill 1& \hfill 12\\ \hfill 3& \hfill 4& \hfill 8& \hfill -3\end{array}\right]\sim \left[\begin{array}{cccc}\hfill 1& \hfill 2& \hfill 6& \hfill -5\\ \hfill 0& \hfill 1& \hfill 5& \hfill -6\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right]$
   2. $A=\left[\begin{array}{cccc}\hfill -3& \hfill 9& \hfill -2& \hfill -7\\ \hfill 2& \hfill -6& \hfill 4& \hfill 8\\ \hfill 3& \hfill -9& \hfill -2& \hfill 2\end{array}\right]\sim \left[\begin{array}{cccc}\hfill 1& \hfill -3& \hfill 6& \hfill 9\\ \hfill 0& \hfill 0& \hfill 4& \hfill 5\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right]$
   3. $A=\left[\begin{array}{ccccc}\hfill 1& \hfill 4& \hfill 8& \hfill -3& \hfill -7\\ \hfill -1& \hfill 2& \hfill 7& \hfill 3& \hfill 4\\ \hfill -2& \hfill 2& \hfill 9& \hfill 5& \hfill 5\\ \hfill 3& \hfill 6& \hfill 9& \hfill -5& \hfill -2\end{array}\right]\sim \left[\begin{array}{ccccc}\hfill 1& \hfill 4& \hfill 8& \hfill 0& \hfill 5\\ \hfill 0& \hfill 2& \hfill 5& \hfill 0& \hfill -1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 4\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right]$

3. [2.8.{21,22}] True/False. Justify your answers.

   1. A subspace of ${ℝ}^n$ is any set $H$ such that (i) the zero vector is in $H$, (ii) $𝐮,𝐯$, and $𝐮+𝐯$ are in $H$, and (iii) $c$ is a scalar and $c𝐮$ is in $H$.
   2. If ${𝐯}_1,\dots ,{𝐯}_p$ are in ${ℝ}^n$, then $\mathrm{Span}\{{𝐯}_1,\dots ,{𝐯}_p\}$ is the same as the column space of the matrix $[{𝐯}_1\cdots {𝐯}_p]$.
   3. The set of all solutions of a system of $m$ homogeneous equations in $n$ unknowns is a subspace of ${ℝ}^m$.
   4. The columns of an invertible $n\times n$ matrix form a basis for ${ℝ}^n$.
   5. Row operations do not affect linear dependence relations among the columns of a matrix.
   6. A subset $H$ of ${ℝ}^n$ is a subspace if the zero vector is in $H$.
   7. Given vectors ${𝐯}_1,\dots ,{𝐯}_p$ in ${ℝ}^n$, the set of all linear combinations of these vectors is a subspace in ${ℝ}^n$.
   8. The null space of an $m\times n$ matrix is a subspace of ${ℝ}^n$.
   9. The column space of a matrix $A$ is the set of solutions of $A𝐱=𝐛$.
   10. If $B$ is an echelon form of a matrix $A$, then the pivot columns of $B$ form a basis for $A$.