mma2.txt Mathematica Session 2 Math 1b January 16, 2009
On the next page, we calculate the inverse of a "random" 5 by 5 matrix. On the following pages, we try to find a nonnegative solution to a system of linear equations Ax = b . We fail. But by carrying
Math 1b Types of linear combinations A vector u is a linear combination of vectors v1 , v2 , . . . , vk when there are scalars c1 , c2 , . . . , ck so that u = c1 v1 + c2 v2 + . . . + ck vk . A nonempty set U of vectors is a subspace (or linear subspace)
Math 1b Practical February 26, 2009 Help with Mathematica for Problem Set 7 Remember that Mathematica works with matrices as a "list" of rows, each row being a "list". So the matrix A = 12 34 is entered and diplayed as cfw_1,2,cfw_3,4. Use a dot (period)
Math 1b practical January 28, 2009 Mathematica Session 3 (Are these "sessions" of any value?) At In[1], we teach Mathematica how to produce "random" m by n matrices. At In[2], we teach Mathematica an abbreviation for MatrixForm. At In[3], we construct a 5
Math 1b A story where a nonnegative solution is required January 16, 2009 An ounce of food substances F1 , F2 , F3 , F4 provides the following percentages of a students minimum daily requirements of three nutrients: carbohydrates, vitamin X, and mineral Y
We use matrices to model systems of linear equations, both homogeneous and inhomogeneous. 2x x+ 2x x+ 3y y 3y y + +z 5z z 5z = = = = 7 3 0 0 2 1 3 1 1 5 17 5 3
2 3 11
Rather than work with systems of equations, we prefer to work with the the corresponding
1 Math 1b Practical Eigenvalues and diagonalization February 17, 2009 In certain applications, one is interested in what happens when the same square matrix is applied to a vector a number of times. Example: the rabbit epidemic. The problem in this exampl
Math 1b Practical Determinants February 11, 2009 Expanded February 18 Square matrices have determinants, which are scalars. Determinants can be introduced in several ways; we choose to give a recursive denition. The determinant of a 1 1 matrix is the entr
Math 1b Practical March 9, 2009 LinearProgramming[c,A,b] solves what I would call the transpose-dual-canonical form LP problem - it finds a vector x that minimizes the value of cx subject to Ax >= b and x >= 0 . In In[1], I define LP[c,A,b] which solves t
1 Math 1b Practical, March 9, 2009 Intoduction to Linear Programming.
Duality
Recall the primal and dual canonical forms for LP problems:
(PRIMAL) CANONICAL: DUAL CANONICAL:
maximize cx subject to minimize yb subject to
Ax b and x 0. yA c and y 0.
(1)
Her
1 Math 1b Practical, March 6, 2009 Intoduction to Linear Programming Part 1 A linear programming problem (LP problem) is one that asks for the minimum or maximum of some linear function (the objective function) of variables x1 , x2 , . . . , xn over a dom
1 Math 1b Practical Isometries March 4, 2009 We have explained that a linear transformation of Rn to itself is an isometry when its matrix is an orthogonal matrix. [The converse is also true.] An example of an isometry is the reection through a subspace U
Math 1b Practical Determinants February 11, 2009 Square matrices have determinants, which are scalars. Determinants can be introduced in several ways; we choose to give a recursive denition. The determinant of a 1 1 matrix is the entry of the matrix. Once
Math 1b Prac Orthogonal bases, orthogonal projection January 29, 2009 [This is an outline of material I want to cover before starting Chapter 4 of the text. Examples and motivation will be given in class.] We will write x, y for the dot product of vectors