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# Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Office 432 Carver Phone 515-294-8141 E-mail: hent - PowerPoint PPT Presentation

Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Office 432 Carver Phone 515-294-8141 E-mail: [email protected] http://www.math.iastate.edu/hentzel/class.307.ICN Text: Linear Algebra With Applications, Second Edition Otto Bretscher.

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Spring, 2003

Hentzel

Time: 1:10-2:00 MWF

Room: 1324 Howe Hall

Office 432 Carver

Phone 515-294-8141

E-mail: [email protected]

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

Text: Linear Algebra With Applications,

Second Edition Otto Bretscher

No hand-in-homework assignment

Main Idea: I do not want any surprises on the test.

Key Words: Practice test

Goal: Test over the material taught in class.

|y| |y-x|

linear transformation.

True. It has matrix | 1 -1 |.

|-1 1 |

• 4. The formula (A2)-1 = (A-1)2 holds

• for all invertible matrices A.

• True. A A A-1 A-1 = I.

• | 1 1 1 | be a 5x3 matrix.

• Matrix | 1 0 1 | is invertible.

• | 1 1 0 |

• True.

• | 1 1 1 | | 1 0 1| | 1 0 0 |

• | 1 0 1 | ~ | 0 1 0| ~ | 0 1 0 |

• | 1 1 0 | | 0 1 -1| | 0 0 1 |

• 14. The function be a 5x3 matrix.

• T|x| = |(y+1)2 – (y-1)2 | is a linear

• |y| |(x-3)2 – (x+3)2 |

• transformation.

• True. T|x| = | 4 y|.

• |y| |-12 x|

• 15. Matrix | k -2 | is invertible for all be a 5x3 matrix.

• | 5 k-6 |

• real numbers k.

• True.

• | k -2 | ~ | 1 (k-6)/5 | ~ | 1 (k-6)/5 |

• | 5 k-6 | | k -2 | | 0 (-k^2+6k-10)/5|

• This polynomial has roots 3 (+/-) i so for all REAL numbers k, the RCF is I and it is invertible.

• 17. There is a real number k such that be a 5x3 matrix.

• the matrix | k-2 3 | fails to be

• | -3 k-2 |

• invertible.

• False.

• | k-2 3 | ~ | 1 -(k-2)/3 | ~ | 1 -(k-2)/3 |

• | -3 k-2 | | k-2 3 | | 0 (k-2)2+3|

• the roots are k = 2 (+/-) i Sqrt[3] which are

• not real.

• 20. There is a matrix A such that be a 5x3 matrix.

• | 1 2 | A | 5 6 | = | 1 1 |.

• | 3 4 | | 7 8 | | 1 1 |

• True | 1 2 | -1 | 1 1 | | 5 6 ||-1

• | 3 4 | | 1 1 | | 7 8 |

• Should work. 1/2 | 1 -1 |

• | -1 1 |

• 21. There is a matrix A such that be a 5x3 matrix.

• A | 1 1 | = | 1 2 |.

• | 1 1 | | 1 2 |

• False Any linear combination of the rows

• of | 1 1 | will look like | x x |.

• | 1 1 | | y y |

• 24. | 1 k | be a 5x3 matrix.3 = | 1 3k | for all real

• | 0 1 | | 0 1 |

• numbers k.

• True:

• 25. The matrix product be a 5x3 matrix.

• | a b | | d -b | is always a scalar

• | c d | | -c a |

• of I2.

• True. The scalar is ad-bc.

• 30. If A rows.2 = In, then matrix A must be invertible.

• True. In fact, A is its own inverse.

• 31. If A rows.17 = I2, then A must be I2.

• False A = | Cos[t] -Sin[t] |

• | Sin[t] Cos[t] |

• Where t = 2 Pi/17 should work.

• 32. If A rows.2 = I2 , then A must be either I2 or –I2.

• False A = | -1 0 | is one possibility.

• | 0 1 |

• 36. If A rows.2 = A for an invertible nxn matrix

• A, then A must be In.

• True. Multiply through by A-1 giving A=I.

• 38. The equation A rows.2 = A holds for all 2x2

• matrices A representing an orthogonal

• projection.

• True. Once you have projected once by

• A, subequent actions by A will simply fix the

• vector.

• 40. If A rows.2 is invertible, then

• matrix A itself must be invertible.

• True. For A2 to be defined, then

• A must be square. If AAB = I, then

• A must be right invertible so A is

• invertible.

• 41. The equation A rows.-1 = A holds for all 2x2

• matrices A representing a reflection.

• True. For a reflection A2 = I.

• 45. If A rows.2 + 3A + 4 I3 = 0 for a 3x3 matrix

• A then A must be invertible.

• True. A(A+3) = -4 I3

• so the inverse of A is (-1/4)(A+3).

• 48. If T is any linear transformation rows.

• from R3 to R3, then T(VxW) = T(V)xT(W)

• for all vectors V and W in R3.

• | 1 0 1 | | 1 | | 0 |

• False. T = | 0 1 1 | V = | 0 | W = | 0 |

• | 0 0 1 | | 0 | | 1 |

• | 0 | | 0 | | 1 | | 1 | | 0 |

• T[VxW] = T| -1 | = |-1 | (TV)x(TW) = | 0 | x| 1 } = | -1 |.

• | 0 | | 0 | | 0 | } 1 } | 1 |

• 49. There is an invertible 10x10 matrix rows.

• that has 92 ones among its entries.

• False. There are only 8 entries which

• are not one. At least 2 columns have

• only ones. Matrices with 2 identical

• columns are not invertible.