:: Column space practise problems

1. In each of the following,
   1. determine a basis for the column space of $A$;
   2. express the other columns of $A$ as linear combinations of the basis vectors from part i;
   3. if possible, express $\mathbf{b}$ as a linear combination of the basis vectors from part i; and
   4. determine whether the system $A\mathbf{x}=\mathbf{b}$ is consistent.
   1. $A=\left[\begin{array}{rr}1& 3\\ 4& -6\end{array}\right]$, $\mathbf{b}=\left[\begin{array}{r}-2\\ 10\end{array}\right]$
   2. $A=\left[\begin{array}{ccc}1& 1& 2\\ 1& 0& 1\\ 2& 1& 3\end{array}\right]$, $\mathbf{b}=\left[\begin{array}{r}-1\\ 0\\ 2\end{array}\right]$
   3. $A=\left[\begin{array}{rrr}1& -1& 1\\ 9& 3& 1\\ 1& 1& 1\end{array}\right]$, $\mathbf{b}=\left[\begin{array}{r}5\\ 1\\ -1\end{array}\right]$
   4. $A=\left[\begin{array}{rrr}1& -1& 1\\ 1& 1& -1\\ -1& -1& 1\end{array}\right]$, $\mathbf{b}=\left[\begin{array}{c}2\\ 0\\ 0\end{array}\right]$
   5. $A=\left[\begin{array}{cccc}1& 2& 0& 1\\ 0& 1& 2& 1\\ 1& 2& 1& 3\\ 0& 1& 2& 2\end{array}\right]$, $\mathbf{b}=\left[\begin{array}{c}4\\ 3\\ 5\\ 7\end{array}\right]$
2. In each of the following, a spanning set for a subspace of ${\mathbb{R}}^4$ is given. Reduce the spanning set to a basis for the subspace; that is, determine a subset of the spanning set that is a basis for the subspace.
   1. ${\mathbf{v}}_1=(1,0,1,1)$, ${\mathbf{v}}_2=(-3,3,7,1)$, ${\mathbf{v}}_3=(-1,3,9,3)$, ${\mathbf{v}}_4=(-5,3,5,-1)$.
   2. ${\mathbf{v}}_1=(1,-2,0,3)$, ${\mathbf{v}}_2=(2,-4,0,6)$, ${\mathbf{v}}_3=(-1,1,2,0)$, ${\mathbf{v}}_4=(0,-1,2,3)$.
   3. ${\mathbf{v}}_1=(1,-1,5,2)$, ${\mathbf{v}}_2=(-2,3,1,0)$, ${\mathbf{v}}_3=(4,-5,9,4)$, ${\mathbf{v}}_4=(0,4,2,-3)$, ${\mathbf{v}}_5=(-7,18,2,-8)$.

Answers