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====== Math 445 lab 4: linear algebra and the truss ======

**Problem 1:** Compute the following products in your head or with pencil and paper, and then check your answers with Matlab. Note: one of the problems is not well-posed and therefore does not have an answer. 

**(a)** 
<latex>
x = 
\left[\begin{array}{ccc}
4 & 3 & 8
\end{array} \right]
</latex>
and
<latex>
y = 
\left[\begin{array}{c}
2 \\
1 \\
-4
\end{array} \right]
</latex>.
 What is $xy$?

**(b)** 
<latex>
x = 
\left[\begin{array}{ccc}
9 & 1 & 0
\end{array} \right]
</latex>
and
<latex>
y = 
\left[\begin{array}{ccc}
-5 & 6 & 2
\end{array} \right]
</latex>.
 What is $xy$?

**%%(c)%%**
<latex>
A = \left[
\begin{array}{cc}
3 & 1 \\
2 & -7 
\end{array}
\right]
</latex>
and
<latex>
x = \left[
\begin{array}{cc}
2 \\
1
\end{array}
\right]
</latex>.
 What is $Ax$?

**(d)** 
<latex>
A = \left[
\begin{array}{cc}
3 & 1 \\
2 & -7 
\end{array}
\right]
</latex>
and
<latex>
B = \left[
\begin{array}{cc}
4 & -6 \\
0 & 2
\end{array}
\right]
</latex>.
 What is $AB$?

**Problem 2:** Suggest a slight modification that turns the ill-posed problem from **1** into a well-posed problem, and then compute it by hand and with Matlab.

**Problem 3:** Given the 2 x 2 matrix 
<latex>
A = \left[
\begin{array}{cc}
4 & 2 \\
-1 & 5 
\end{array}
\right]
</latex>
and the 2-d vector
<latex>
b = \left[
\begin{array}{cc}
3 \\
4
\end{array}
\right]
</latex>, 
 what vector $x$ satisfies $Ax=b$? Solve with pencil and paper and with Matlab. 


**Problem 4:** Solve the following systems of equations numerically with Matlab.

**(a)** 
<latex> \quad
\begin{align*}
2x -y &= 8 \\
6x-5y &= 32 \\
\end{align*}
</latex>


**(b)** 
<latex> \quad
\begin{align*}
y + z &= 6 \\
3x-y+z &= -7 \\
x+y-3z &= -13
\end{align*}
</latex>

**%%(c)%%**
<latex> \quad
\begin{align*}
2x + y - 3z &= 0 \\
6x +3y -8zz &= 0 \\
2x -y + 5z &= -4
\end{align*}
</latex>

**Problem 5:** Use Matlab to solve the problem. Nilanjana has 40 coins worth %%$%%6.40. They're all quarters and nickels. How many nickels and how many quarters does she have? Verify that your answer solves the problem. 

Hint: convert the story problem to an $Ax=b$ problem and then solve that with Matlab. 

**Problem 6:** Use Matlab to solve the problem. Suhasini has 44 coins worth %%$%%7.50. They're all quarter, dimes, and nickels. She has twice as many dimes as nickels. How many of each type of coin does she have? Find the answer, and then verify that the solution satisfies the problem. 

**Problem 7:** Use Matlab to solve the following problem (from //Numerical Computing with Matlab// by Clive Moler).

{{:gibson:teaching:fall-2012:math445:truss2.png?direct&600}}

{{:gibson:teaching:fall-2012:math445:truss3.png?direct&600}}

{{:gibson:teaching:fall-2012:math445:truss4.png?direct&600}}

One of the main challenges of this problem is figuring out how to enter the elements of the matrix quickly and reliably. For a small matrix, it's easy to type out the matrix explicitly like ''A = [ 1 9 7 ; 4 3 8 ; 6 2 0]'', but for a 13 x 13 matrix with 169 elements, that's tedious and error prone.
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