======Math 445 sample exam 1 with solutions ======

**Problem 1.** Write one line of Matlab code that assigns a 3-d column vector with 
components 4,5,7 to variable ''x''.

  x = [4; 5; 7]

or
 
<code>
x = [4 5 7]'
</code>


**Problem 2.**  Write one line of Matlab code that assigns a 3-d row vector with 
components 4,5,7 to variable ''x''.

  x = [4 5 7]


**Problem 3.** **(Note: We haven't done permutations yet so you're not required
to know this material.)** Write Matlab code that simulates the shuffling of a deck 
of cards by producing a random permutation of the integers 1 through 52.

  randperm(52)

**Problem 4.** Write Matlab code that draws a unit circle, using the formulae $x = \cos \theta$
and $y = \sin \theta$ for 200 evenly spaced values of theta between 0 and 2pi. Label the 
axes and make the circle red.

  theta = linspace(0,2*pi, 200);
  x = cos(theta);
  y = sin(theta);
  plot(x,y,'r')
  xlabel('x');
  ylabel('y');
  
or

  theta = linspace(0,2*pi, 200);
  plot(cos(theta), sin(theta), 'r')
  xlabel('x');
  ylabel('y');


**Problem 5.** Write a conditional expression that evaluates to 1 (true) if ''x'' and ''y''
are equal or if either is zero. 


**Problem 6.** Show how to solve the system of equations with three lines of Matlab code.

<latex>
\begin{align*}
2x + 3y - 8 &= 0 \\
y - 4z  + 10 &= 0 \\
5z - 2x - 13 &= 0
\end{align*}
</latex>

Note that the above equations need to be rearranged first

<latex>
\begin{align*}
2x + 3y + 0z &= 8 \\
0x +  y - 4z &= -10 \\
-2x + 0y + 5z &= 13
\end{align*}
</latex>

That can now be written as an ''Ax=b'' problem

<latex>
\begin{align*}
\left( \begin{array}{rrr}
      2 & 3 & 0 \\
      0 & 1 & -4 \\
      -2 & 0 & 5 
\end{array} \right)
\left( \begin{array}{rrr}
  x \\ y \\ z
\end{array} \right)
= 
\left( \begin{array}{rrr}
  8 \\ -10 \\ 13
\end{array} \right)
\end{align*}
</latex>

To solve in Matlab, 

  A = [2 3 0; 0 1 -4; -2 0 5];
  b = [8; -10; 13];
  x = A\b


**Problem 7** Let ''A'' be an ''M x K'' matrix and ''B'' be an ''K x N'' matrix. Then the 
product ''C = AB'' is an ''M x N'' matrix whose elements are given by

<latex>
\begin{align*}
  C_{ij} = \sum_{k=1}^{K} A_{ik} B_{kj}
\end{align*}
</latex>

Write a Matlab function ''matmatmult'' that returns the product ''C'' of matrices 
''A'' and ''B''. Use the above formula instead of Matlab's built-in matrix 
multiplication!

 
<code>
function C = matmatmult(A,B);
% compute matrix-matrix product C = A*B

  % find sizes of matrices
  [M,K] = size(A);  
  [K,N] = size(B);  
  
  % allocate M x N matrix for answer C
  C = zeros(M,N);

  % compute each element C(i,j)...
  for i=1:M
    for j=1:N
      
      % ...as sum from 1 to k of A(i,k)*B(k,j)
      for k=1:K
        C(i,j) = C(i,j) + A(i,k)*B(k,j);
      end

    end
  end
  
end
</code>



**Problem 8.** Matrix multiplication ''C = AB'' is defined only for compatible matrices:
the number of columns of ''A'' must equal the number of rows of ''B''. Write a short piece of 
Matlab code that could be inserted in your ''matmatmult'' function that prints an error message
and returns a null (''0 x 0'') matrix if ''A'' and ''B'' are incompatible. 

I'll provide the whole function with the inserted code
<code>
function C = matmatmult(A,B);
% compute matrix-matrix product C = A*B

  % find sizes of matrices
  [M,K1] = size(A);  
  [K2,N] = size(B);  
  
  % test that # cols of A == # rows of B
  if K1 ~= K2 
    fprintf('matmatmult dimension mismatch: A has %d cols but B has %d rows\n', K1, K2);

    % reset M,N to zero so that the function returns a 0 x 0 value for C
    M = 0;  
    N = 0;
  end

  % allocate M x N matrix for answer C
  C = zeros(M,N);

  % compute each element C(i,j)...
  for i=1:M
    for j=1:N
      
      % ...as sum from 1 to k of A(i,k)*B(k,j)
      for k=1:K
        C(i,j) = C(i,j) + A(i,k)*B(k,j);
      end

    end
  end
  
end
</code>