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
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)
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
Solutions to Homework 6
(Note: some computation details omitted)
Problem 2.
(a) The matrix is upper triangular, so its eigenvalues are just the entries on its main diagonal: 2, 5, 4.
(b) The matrix is diagonalizable, as the eigenvalues are distinct (Corol
Solutions to Homework 2
Problem 2.
Let C be the inverse of AB. Then I = C(AB) = (CA)B, and so B is left invertible, with left inverse
CA. Likewise, I = (AB)C = A(BC), so A is right invertible, with right inverse BC.
Problem 3.
Let cfw_ei 4i=1 denote the s
Solutions to Homework 4
Problem 2.
Let Ax = b be such a system. Then Ax = 0 must have solution set s(1, 2, 1)T . Note that there is
exactly one free variable; one possibility for A is
1 0 1
.
0 1 2
Now (1, 1, 0)T is a solution to Ax = b, so for this choic
Solutions to Homework 3
Problem 2. (Steps omitted)
19 5 1
1
6 2
0
2
5 1
1
3
3 0
1
3 1 2
6
0 1 1
Problem 3.
Define the isomorphism T : V F3 where T (u) = e3 , T (v) = e2 , T (w) = e1 . Then w, v + w, and
u + v + w form a basis for V iff their images under
Solutions to Homework 1
Problem 2.
1. Not a subspacethe subset contains v = (1, 1, . . . , 1)T but not 2v = (2, 2, . . . , 2)T .
2. This is a subspace of Fn . Take any two members, x = (x1 , . . . , xn )T and y = (y1 , . . . , yn )T , and
any c F. Then cx
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
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
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
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, 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
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
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
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
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