Math 307 Spring, 2005 Hentzel Time: 10:00 to 10:50 MWF Room: 205 Carver Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141 E-mail: hentzel@iastate.edu http://orion.math.iastate.edu/hentzel/class.307.05 Text: Linear Algebra and its Applications, Third Edition David C. Lay Friday, January 14 Chapter 1.3 Page 47 Problems 11, 25, 29, 30, 31, 33 Main Idea: How to add and multiply matrices. Key Words: Linear Combination, AX=B, Row, Column, i-j Entry, Rank Goal: Understand the interplay of matrix multiplication in terms of linear combinations of the rows, or the columns. -------------------------------------------------------------- Previous Assignment Page 37 Problem 11 Determine if b is a linear combination of a1,a2,a3 a1 a1 a3 | 1| |0| | 5| | 2 | x | -2|+y |1|+z |-6| = | -1 | | 0| |2| | 8| | 6 | | 1 0 5 2 | | 0 1 4 3 | | 0 2 8 6 | a | 1 0 5 2 | | 0 1 4 3 | | 0 0 0 0 | | x | | 2 | | -5 | | y | = | 3 |+a | -4 | | z | | 0 | | 1 | Page 37 Problem 27 A mining company has two mines. Copper Silver Mine 1 20 550 Mine 2 30 500 a. What physical interpretation is 5 (20,550) ? b. How to produce exactly 150 copper and 2825 silver. x| 20 | + y | 30 | = | 150 | | 550 | |500 | | 2825 | 1 0 3/2 0 1 4 Mine one works a day and a half and mine 2 works 4 days. RowReduce[ {{20,30,150},{550,500,2825}}]; Page 37 Problem 28 A B heat 27.6 30.2 S02 3100 6400 soot 250 360 (a) How much heat does the steam plant produce when it burns x1 tons of A and x2 tons of B. 27.6 x1 + 30.2 x2 (b) In general | heat | | 27.6 | | 30.2 | | S02 | = x1 | 3100 |+ x2 | 6400 | | soot | | 250 | | 360 | (c) | 162 | | 27.6 | | 30.2 | | 23610 | = x1 | 3100 |+ x2 | 6400 | | 1623 | | 250 | | 360 | RowReduce[ {{27.6,30.2,162},{3100,6400,23610},{250,360,1623}} ]; 1 0. 3.9 0 1 1.8 0 0 0 Solution: 3.9 A and 1.8 B. --------------------------------------------------------------- New Material: A matrix is a rectangular array of numbers. (a) To multiply a matrix by a number, multiply every element of the matrix by that number. | 1 2 1 | | 3 6 3 | 3 | 2 1 0 | = | 6 3 0 | | 0 0 1 | | 0 0 3 | | 2 2 2 | | 6 6 6 | ------------------------------------------------------------- (b) To add two matrices together, simply add the corresponding entries together. The matrices must have the same size so that the elements will correspond. | 1 2 1 | + | 3 0 1 | = | 4 2 2 | | 3 0 -1 | | 2 1 0 | | 5 1 -1 | -------------------------------------------------------------- (c) Matrix multiplication is more difficult. The reason it is defined as such, is because this combination of the numbers is needed again and again. The real surprise is that multiplication is associative. That i.e. (AB)C = A(BC). | 2 | Row x column [ 1 2 3 ] | 1 | = 1x2 + 2x1 + 3x(-1) = 1 |-1 | ------------------------------------------------------------- And all other products are built up from the basic row times a column. Simply take each and every row times each and every column and arrange the products in a matrix. | 2 3 | [1 2 3] | 1 2 | = [ 1 7] |-1 0 | | 0 2 3 | [1 2 3] | 1 0 2 | = [ 5 -1 10 ] | 1 -1 1 | | 1 1 1 | | 1 0 | | 2 2 | | 3 0 1 | | 0 1 | = | 4 1 | | 0 1 2 | | 1 1 | | 2 3 | | 1 0 1 | | 2 1 | ------------------------------------------------------------- In the product AB, the number of rows of A determines the number of rows of the product. The number of columns of B determines the number of columns in the product. A B = C rs st rt The inner subscripts have to be equal so that the rows and columns are the same length. Note: To add matrices, they must be of the exact same size. A + B = C rs rs rs To multiply matrices, the innermost subscripts must agree. A * B = C rs st rt ------------------------------------------------------------ In a matrix theory course, we only present material that has wide spread applications. We neglect the rest. What is neglected is left undefined. Fast Matrix Multiplication: | 0 1 0 0 | | 2 1 3 | | 1 3 5 | | 0 0 0 1 | | 1 3 5 | = | 9 1 3 | | 1 0 0 1 | | 7 1 2 | |11 2 9 | | 9 1 3 | -------------------------------------------------------------- | 8 2 3| | 1 0 | | 11 5 | | 7 3 1| | 0 1 | = | 8 4 | | 2 8 l| | 1 1 | | 3 9 | | 1 1 9| | 10 10 | | 9 0 2| | 11 2 | | 3 2 1| | 4 3 | ------------------------------------------------------------ | 1 0 1 | | 1 0 2 | | 1 1 1 | | 0 1 1 | | 1 2 0 | = | 1 3 -1 | | 1 0 0 | | 0 1-1 | | 1 0 2 | ---------------------------------------------------------- A linear combination of elements X1,X2, ... Xn with coefficients c1 c2 ... cn is the element c1 X1 + c2 X2 + ... + cn Xn. The rows of AB are linear combinations of the rows of B with coefficients from the rows of A. The columns of AB are linear combinations of the columns of A with coefficients from the columns of B. ------------------------------------------------------------ The system of linear equations; 3x + 2 y + 3 z = 5 2x - y + z = 8 x - 3 y + 4 z = 12 can be thought of finding the coefficients x,y,z which demonstrate that | 5 | | 3 | | 2 | | 3 | | 8 | is a linear combination of | 2 | |-1 | | 1 | |12 | | 1 | |-3 | | 4 | That is, find x,y,z such that | 3 | | 2 | | 3 | | 5 | x| 2 |+y|-1 |+z| 1 | = | 8 | | 1 | |-3 | | 4 | |12 | ------------------------------------------------------------------- | 3 2 3 5 | | 2 -1 1 8 | | 1 -3 4 12 | | 1 0 0 81/32 | | 0 1 0 -73/32 | | 0 0 1 21/32 | | 3 | | 2 | | 3 | | 5 | 81/32| 2 |-73/32|-1 |+21/32| 1 | = | 8 | | 1 | |-3 | | 4 | |12 | ----------------------------------------------------------- A useful work is the "Rank of a Matrix". For now, the rank of a matrix is the number of nonzero rows in the row canonical form of the matrix. (a) The rank makes sense because the Row Canonical Form is unique. But right now it sure is not clear that there are not more than one way to put a matrix into row canonical form. (b) For the simultaneous linear equations AX = B. (i) There is no solution if rank A < rank[A|B]. (ii) if rank A = rank[A|B] then (*) There is a unique solution if rank A = the number of columns of A. (**) There is an infinite number of solutions if rank A < the number of columns of A.