Math 307 Spring, 2005 Hentzel Time: 10:00 to 10:40 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 Monday, February 14 2.7 Main Idea: Spacial rotations about an Axis Key Words: Quaternions Goal: Be able to set up a 3x3 matrix which does a rotation about an axis. Previous Assignment Page 156 Problems 1,2,3,4,9 Page 156 Problem 1 Manufacturing Agriculture Services .10 .60 .60 .30 .20 0 .30 .10 .10 What are the intermediate demands if Agriculture plans to produce 100 units. | 60 | | 20 | | 10 | ---------------------------------------- Page 156 Problem 2 Determine the production levels needed to satisfy a final demand of 18 units for agriculture, with no final demand for the other sectors. Do not compute an inverse matrix. A = {{.10, .60, .60}, {.30, .20, 0}, {.30, .10, .10}}; B = IdentityMatrix[3]; Do[ B = IdentityMatrix[3] + B.A,{i,1,100}]; -1| 0| | 2.22222 1.85185 1.48148 | 0 | |100/3| (I-A) |18| = | 0.833333 1.94444 0.555556 | 18 | = | 35 | | 0| | 0.833333 0.833333 1.66667 | 0 | | 15 | |100/3| | 100/3 | A.| 35 | = | 17 | | 15 | | 15 | ------------------------------------------ Page 156 Problem 3 Determine the production levels needed to satisfy a final demand of 18 units for manufacturing, with no final demand for the other sectors. (Do not compute an inverse matrix) -1|18| | 2.22222 1.85185 1.48148 | 18 | | 40 | (I-A) | 0| = | 0.833333 1.94444 0.555556 | 0 | = | 15 | | 0| | 0.833333 0.833333 1.66667 | 0 | | 15 | |40| |22| A.|15| = |15| |15| |15| Page 156 Problem 4 Determine the production levels needed to satisfy a final demand of 18 units for manufacturing, 18 units for agriculture, and 0 units for services. -1|18| | 2.22222 1.85185 1.48148 | 18 | |220/3 | (I-A) |18| = | 0.833333 1.94444 0.555556 | 18 | = | 50 | | 0| | 0.833333 0.833333 1.66667 | 0 | | 30 | |220/3 | |166/3 | A.| 50 | = | 32 | | 30 | | 30 | --------------------------------------------- Page 156 Problem 9 Solve the Leontief production equation for an economy with three sectors given that | 0.2 0.2 0 | | 40 | C = | 0.3 0.1 0.3 | D =| 60 | | 0.1 0 0.2 | | 80 | A = {{ 0.2, 0.2, 0 }, { 0.3, 0.1,0.3 }, { 0.1, 0, 0.2 }}; d = {40,60,80}; ans = Inverse[IdentityMatrix[3]-A].d | 82.7586 | ans = | 131.034 | | 110.345 | -------------------------------------------------------------- Previous Assignment The Leontief Problem The production of the plants R, S, and T for some period of time is given below. R S T Consumer Total R 10 10 30 30 80 S 10 20 10 20 60 T 60 20 10 10 100 The Leontief input-output model for the open model is: X = AX + D. (1) What is the matrix A? | 145 | (2) Solve the equation X = AX+D for X when D = | 290 | | 145 | A | 10/80 10/60 30/100 | | 10/80 20/60 10/100 | | 60/80 20/60 10/100 | Check with original system: A X + D = X |10/80 10/60 30/100|| 80| |30| | 80| |10/80 20/60 10/100|| 60| + |20| = | 60| It checks. |60/80 20/60 10/100||100| |10| |100| | 145 | Calculating X the new system where consumer demand is | 290 | | 145 | AX+D=X D = (I-A)X (I-A)^(-1) D = X | |-1 | | 1 0 0 | |10/80 10/60 30/100| | | 145 | | 616 | X = | | 0 1 0 | - |10/80 20/60 10/100| | | 290 | = | 690 | | | 0 0 1 | |60/80 20/60 10/100| | | 145 | | 930 | | | -1 | 7 1 3 | | - -(-) -(--) | | 8 6 10 | | | | 145 | | 1 2 1 | | 290 | X = | -(-) - -(--) | | 145 | | 8 3 10 | | | | 3 1 9 | | -(-) -(-) -- | | 4 3 10 | | 272 24 104 | | --- -- --- | | 145 29 145 | | | | 18 54 12 | | 145 | X = | -- -- -- | | 290 | | 29 29 29 | | 145 | | | | 52 40 54 | | -- -- -- | | 29 29 29 | | 616 | X = | 690 | | 930 | --------------------------------------------------- Rationals c Reals c Complexes c Quaternions c Octonions c Sedenions. not not not not much ordered commutative associative at all The quaternions are defined by quaternions = {a+bi+cj+dk} where i^2 = j^2 = k^2 = -1 and i clockwise is positive. counter clockwise is negative. k j The norm is defined by 2 2 2 2 n(a+bi+cj+dk) = a +b +c +d . and the norm is multiplicative. n[x[{a,b,c,d},{p,q,r,s}]] 2 2 (d p - c q + b r + a s) + (c p + d q + a r - b s) + 2 2 (b p + a q - d r + c s) + (a p - b q - c r - d s) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = a p + b p + c p + d p + a q + b q + c q + d q + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 a r + b r + c r + d r + a s + b s + c s + d s 2 2 2 2 2 2 2 2 = (a + b + c + d ) (p + q + r + s ) = n(a,b,c,d) n(p,q,r,s). The conjugate of a+bi+cj+dk = a-bi-cj-dk and conjugation is multiplicative as well. _ Also x x = n(x). _ The quaternions have division. 1/x = x/n(x). The nicest thing about the quaternions is that they represent rotations. -1 If n(x) = 1, then x(0,a,b,c) x rotates the vector (a,b,c). The axis of rotation is the i,j,k coordinates of x. The amount of rotation is 2 ArcCos of the real part of x. Assignment: 1. Find a quaternion which rotates through 30 degrees about the axis vector <1,2,3>. 2. Find a 3x3 matrix which rotates through 30 degrees about the axis vector <1,2,3>.