Linear Algebra (Math 220)
Assignment due Tuesday, February 26

1.  Reading

Relevant Reading

• Lay § 4.2

• Hefferon §§ 3.II – 3.III

• Matthews §§ 3.1 – 3.2

2.  Exercises

1. Let $C$ be the $4\times 4$ matrix $$\left(\begin{array}{cccc}\hfill 1& \hfill 2& \hfill 0& \hfill 2\\ \hfill -2& \hfill -1& \hfill 3& \hfill 2\\ \hfill -2& \hfill 2& \hfill 6& \hfill -1\\ \hfill 1& \hfill 0& \hfill -2& \hfill 0\end{array}\right)\text{,}$$ and let $f$ be the linear map (or function) from ${\mathbf{R}}^4$ to ${\mathbf{R}}^4$ defined by the formula $$y=f\left(x\right)=Cx\text{.}$$

   1. Find all solutions of $f\left(x\right)=\left(0,0,0,0\right)$.

   2. Find all solutions of $f\left(x\right)=\left(1,-2,-2,1\right)$ with $x_3=0$.

   3. Find all solutions of $f\left(x\right)=\left(1,-2,-2,1\right)$.

   4. Find all solutions of $f\left(x\right)=\left(-1,-7,2,1\right)$ with $x_3=0$.

   5. Find all solutions of $f\left(x\right)=\left(-1,-7,2,1\right)$.

   6. What is the kernel of $f$ ?

   7. Find equations that characterize the image of $f$.

2. Let $G$ be the $4\times 4$ matrix $$\left(\begin{array}{cccc}\hfill 1& \hfill 2& \hfill 0& \hfill 1\\ \hfill -2& \hfill -1& \hfill 1& \hfill 1\\ \hfill -1& \hfill 4& \hfill 2& \hfill 5\\ \hfill 5& \hfill 7& \hfill -1& \hfill 2\end{array}\right)\text{,}$$ and let $g$ be the linear map (or function) from ${\mathbf{R}}^4$ to ${\mathbf{R}}^4$ defined by the formula $$y=g\left(x\right)=Gx\text{.}$$ Solve each of the following systems of $4$ linear equations in $4$ unknowns $x_1,x_2,x_3$ and $x_4$.

   1. $g\left(x\right)=\left(0,0,0,0\right)$.

   2. $g\left(x\right)=\left(1,-1,1,3\right)$ with $x_3=0$.

   3. $g\left(x\right)=\left(1,-1,1,4\right)$ with $x_3=0$.

   4. $g\left(x\right)=\left(1,-1,1,4\right)$ with $x_3=x_4=0$.

   5. $g\left(x\right)=\left(3,-1,2,1\right)$ with $x_3=0$.

   6. $g\left(x\right)=\left(3,-1,7,10\right)$ with $x_3=0$.

   7. What is the kernel of $g$ ?

   8. Find equations that characterize the image of $f$.

3. Let $M$ be an $m\times n$ matrix, and let $\phi \left(x\right)=Mx$. Let $a$ and $b$ be any two points of ${\mathbf{R}}^n$. Show that $\phi \left(a\right)=\phi \left(b\right)$ if and only if $a-b$ lies in the kernel of $\phi$.