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 7 2.4 Assignment Page 139 Problem 10, 13, 15, 19 Main Idea: If the individual multiplications are valid, then the partition works. Key Words: Partitioned matrix. Goal: Simplify multiplication by using block structure. Previous Assignment Page 132 Problem 7, 8, 11, 12, 33 Page 132 Problem 7 | -1 -3 0 1 | | 3 5 8 -3 | | -2 -6 3 2 | | 0 -1 2 1 | | 1 3 0 -1 | | 3 5 8 -3 | | -2 -6 3 2 | | 0 -1 2 1 | | 1 3 0 -1 | | 0 -4 8 0 | | 0 0 3 0 | | 0 -1 2 1 | | 1 3 0 -1 | | 0 1 -2 0 | | 0 0 3 0 | | 0 -1 2 1 | | 1 3 0 -1 | | 0 1 -2 0 | It is invertible | 0 0 3 0 | | 0 0 0 1 | Page 132 Problem 8 | 1 3 7 4 | | 0 5 9 6 | It is invertible | 0 0 2 8 | | 0 0 0 10 | Page 132 Problem 11 For a SQUARE matrix A. (a) If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the nxn identity matrix. n (b) If the columns of A span R , then the columns are linearly independent. (c) If A is an nxn matrix, n then the equation Ax=B has at least one solution for each b in R (d) If the equation AX = 0 has a nontrivial solution then A has fewer than n pivot positions. T (e) If A s not invertible Then A is not invertible. Page 132 Problem 12 (a) If there is an nxn matrix D such that AD = I then there is also an nxn matrix C such that CA = I (b) If the columns of A are linearly independent n then the columns of A span R n (c) If the equation AX = B has at least one solution for each B in R then the solution is unique for each B. n n (d) If the linear transformation X-->AX maps R into R then A has n pivot positions. n (e) If there is a B in R such that the equation AX = B is inconsistent, then the transformation X-->AX is not one-to-one. Page 132 Problem 33 -1 Show that T is invertible and find T T(x ,x ) = {-5x +9x , 4x -7x } 1 2 1 2 1 2 | -5 9 || x | = | -5x +9x | | || 1 | | 1 2 | | 4 -7 || x | | 4x -7x | | 2 | | 1 2 | | -7 -9 | | -4 -5 | The inverse of T is ------------ | 7 9 | -1 | 4 5 | New Material: Multiply | I 0 | | A B | = | E I | | C D | Multiply | 0 I | | W X | = | I 0 | | Y Z | Multiply | E 0 | | A B | = | 0 F | | C D | Multiply | X 0 0 | | A Z | | Y 0 I | | 0 0 | | B I | Find the inverse of | A B | | 0 C | Show that the product of two upper triangular matrices remains upper triangular. Solve AX + D = X. In Class Assignment: ------------------------------------------------------------------------------- || || e|| || d || || *********************** **************************** || || f|| || c || || ----------------------- ------------------------------ | * | | * | | * | | ========== * =========== | | a * b | | * | | * | | * | | * | If you measure the cars passing through each of a,b,c,d,e,f can you determine the number of cars that went straight, turned right, or turned left from each approach. NE NW W WS E ES NE + E = c NW + W = e WS + ES = a NE + NW = b W + WS = d E + ES = f ---------------------------------------------------