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'} + c_{2} x, let c = [ c_{1} ; c_{2}] (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.

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