Math 307

Spring, 2003

Hentzel

Time: 1:10-2:00 MWF

Room: 1324 Howe Hall

Instructor: Irvin Roy Hentzel

Office 432 Carver

Phone 515-294-8141

E-mail: hentzel@iastate.edu

http:

//www.math.iastate.edu/hentzel/class.307.ICN

• Text: Linear Algebra With Applications,

• Second Edition, Otto Bretscher

• Wednesday, Feb 5 Chapter 2

Page 94 Problems 1 through 50

• Main Idea: Apply the row operation to the identity.

• Key Words: Differentiation, Tidbits

• Goal: Learn some new tricks.

• Previous Assignment

• Page 85 Problem 34.

• Consider two nxn matrices A and B such that

• the product AB in invertible. Show that the

• matrices A and B are both invertible.

• (AB)-1A B = I

• Since B has a left inverse, B is invertible.

• A B(AB)-1 = I

• Since A has a right inverse, A is invertible.

• Page 85 Problem 50.

• | 1 0 0 |

• Consider the matrix E = |-3 1 0 |

• | 0 0 1 |

• and an arbitrary 3x3 matrix

• | a b c |

• A = | d e f |

• | g h k |

• (a) EA has row two replaced by

• -3 Row 1 + Row 2.

|1 0 0 |

Consider the matrix E = |0 1/4 0 |

|0 0 1 |

• EA multiplies row 2 by 1/4.

• (c) Can you think of a 3x3 matrix E such that EA is obtained from A by swapping the last two rows (for any 3x3 matrix A)?

• | 1 0 0 |

• | 0 0 1 |

• | 0 1 0 |

• (d) The three types of elementary matrices are

• i j• | 1 0 0 ... 0 |• | 0 1 0 ... 0 |• | 0 0 1 ... 0 |• | . . |• i | . 0 1 . | Switch row i and row j.• | . . | • j | 1 0 |• | 1 0|• | 0 0 ... 0 1|

• Multiply row i by c.

• i • | 1 0 0 ... 0|• | 0 1 0 ... 0|• | 0 0 1 ... 0|• | . .|• i| . c 0 .| • | . .|• | 0 1 |• | 1 0|• | 0 0 0 1|

• Add c times row i to row j.

• | 1 0 0 ... 0 |• | 0 1 0 ... 0 |• | 0 0 1 ... 0 |• | . . |• i | . 1 0 . |• | . . |• j | c 1 |• | 1 0 |• | 0 0 ... 0 1 |• i j

• Page 85 Problem 52.

• Justify the following:

• If A is an mxn matrix, then

• there are elementary mxm matrices

• E1 E2, ... Ep

• such that RREF(A) = E1 E2 ... EpA.

• Solution: We can reduce A to row canonical form by using the elementary row operations. Each of these elementary row operations can be viewed as an action performed by mxm matrices multiplying A on the left.

• I would have preferred that the actions start by counting from right to left as

• Ep ... E3 E2 E1 A = RCF(A).

• But the numbering in the book goes the other way. How they know to start with p and work downwards is beyond me.

• Find such elementary matrices E1 E2 ... Ep for

• A = | 0 2 |

• | 1 3 |

E1 = | 0 1 | E1 A = | 1 3 |

| 1 0 | | 0 2 |

E2 = | 1 0 | E2E1A = | 1 3 |

| 0 1/2 | | 0 1 |

• E3 = | 1 -3 | E3E2E1A = | 1 0 |

• | 0 1 | | 0 1 |

Write the matrix of the linear transformation of differentiation with respect to the basis

e 2x , xe2x , x2 e2x , x3 e 2x .

Find some way to use the matrix from part to compute the sixth derivative of

{2 - x + x 2 - x 3} e2x

• e2x x e2x x2 e2x x3 e2x

• e2x 2 0 0 0

• x e2x 1 2 0 0

• x2 e2x 0 2 2 0

• x3 e2x 0 0 3 2

• Take the TRANSPOSE.

• | 2 1 0 0 |

• D = | 0 2 2 0 |

• | 0 0 2 3 |

• | 0 0 0 2 |

0 1 2 3 4 5 6

2 1 0 0 2 3 6 10 0 -96 -544

0 2 2 0 -1 0 -2 -20 -96 -352 -1120

0 0 2 3 1 -1 -8 -28 -80 -208 -512

0 0 0 2 -1 -2 -4 -8 -16 -32 -64

• The sixth derivative is

• e2x (-544 - 1120 x -512 x2 - 64 x3)

(a) The rows of AB are linear combinations of what?

(b) What is the rank of a matrix?

(c) What is the relation between the rank, the nullity, and the number of columns of a matrix?

• (d) What is the relationship between the Row Canonical Form of a matrix and the existence of the inverse of the matrix?

• (e) What is the 2x2 matrix which rotates the plane through 60 degrees?

• (f) Give a non zero 2x2 matrix which is not invertible.

• (g) What are the three elementary row operations.

• (h) When is a function a linear transformation?

• Multiply these two matrices.

| 1 0 0 0 1 0 | | 3 8 2 4 | | 0 1 0 0 0-1 | | 1 0 1 2 | | 0 0 2 0 0 0 | | 3 3 1 2 | | 1 1 1 0 0 0 | | 4 3 1 0 | | 0 0 0 0 1 1 | | 5 4 3 2 | | 1 3 9 2 |

• Solve AX = B and write the answer as • X = X0 + a1X1 + a2X2 + ... arXr and check

• your answer using • A[X0 X1 X2 ... Xr] = [B 0 0 0 ... 0].

• | 1 2 0 1 2 3 | | x | | 2 |

| 1 3 0 0 1 2 | | y | | 1 |

| 2 5 0 1 3 5 | | z | = | 3 |

| 4 10 0 2 6 11 | | w | | 7 |

| u |

| v |

Find the inverse of this matrix.

• | 1 1 2 0 |

• | 1 2 3 0 |

• | 2 0 5 1 |

• | 2 3 4 0 |