if a is an m by n rectangular matrix with m gt n

Question

If A is an m by n rectangular matrix with m >= n and if c is a vector with n components then A c = y can not usually be solved exactly.  Matlab can solve these equations approximately with the command c = A \ y.   The solution c is called the "least squares" solution for reasons that we will describe later in the course.  To illustrate an application involving this use of a rectangular matrix suppose that we wish to fit the three points (0,3),  (2,5), and (4,6) with a straight line.  Then as we described earlier we can let the line be described by c_1' + c₂ x, let c = [ c₁ ; c₂] (using Matlab notation for vectors), let y = the vector of y values.and let A be a Vandermonde matrix with components a_ij= x_I^j-1 .  To fit the line to the data we need to try to solve the equation  A c = y..  In this example one can create A with the Matlab command A = [ 1 0;1 2;1 4] and y with y = [3 ; 5; 6].  Then c = A \ y gives c = [3.1667; .75].  The commands

 x=[0; 2; 4]                 % x isthe vector of x coordinates

 plot(x,y,'o',x,A*c)     % note that A*c generates points on the best fit line

 xlabel('x')

 ylabel('y')

 title('given points (o) and least squares fit (line)')

will plot the given points and the line 3.1667  + .75 x.  See the figure below.

For homework consider the points (0,0), (1,2), (2,3), (3,9), ( 4,17), (5,24), (6,37). (1) Use Matlab to find the least squares best fit with a line.  Turn in A, y and c and a plot like the one above.  Also (2) use Matlab to find the least square best fit with a quadratic.  Turn in the same information.