Least square applications

1)We wish to measure a wavefront u(x,y) on a 3X3 grid (see the attached file). The sensor how ever provides us only with differences between grid points. There is sensor noise, so the difference measurements may not be consistent. Number the grid points linearly and set up a set of linear equations (matrix form) relating the values of u to the values of du. Use least squares to get the best estimate of u(x,y). First use the exact values of du then use du with error, compare. Check by summing du's along a set of defined paths. Does least squares improve accuracy when sensor noise is included? 2)Often we are not interested in the component of u of the form L(x,y)=Ax+By+C use least squa... click for more

Difficulty in setting up the constraints for the problem

The famous Y.S. Chang Restaurant is open 24 hours a day. Waiters and busboys report for duty at 3am, 7am, 11am, 3pm, 7pm or 11pm and each works an 8-hour shift. The following table shows the minimum number of workers needed during the six periods into which the day is divided. The scheduling problem is to determine how many waiters and busboys should report for work at the start of each time period to minimize the total staff required for one day's operation. Period Time # of waiters/busboys required 1 3am-7am 3 2 7am-11am 12 3 11am-3pm 16 4 3pm-7pm 9 5 7pm-11pm 11 6 11pm-3am ... click for more

Linear Algebra -- Linear Transformations

Please help. I always have a hard time with Linear Algebra. What's the difference between mapping from R3 into R2 and mapping from R2 into R3? Why is the following not a linear transformation from R3 into R2? L(x) = (1 + x1, x2)^T And why is this one not a linear transformation from R2 into R3? L(x) = (x1, x2, 1)^T Thanks!

Linear Algebra -- Linear Transformations

Let a be a fixed vector in R2. A mapping of the form L(x) = x+a is called a translation. Show that if a does not equal 0, then L is not a linear transformation. Describe or illustrate geometrically the effect of the translation. Thanks for your help!

Linear Algebra -- Linear Transformations

Determine whether the following are linear transformations from C[0,1] into R^1. L(f) = |f(0)| L(f) = [f(0) + f(1)]/2 L(f) = {integral from 0 to 1 of [f(x)]^2 dx}^(1/2) Thanks so much. :)