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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


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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: [email protected] http://www.math.iastate.edu/hentzel/class.307.ICN Text: Linear Algebra With Applications, Second Edition Otto Bretscher.

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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: [email protected]

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

Text: Linear Algebra With Applications,

Second Edition Otto Bretscher


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Friday, Feb 7 Chapter 2

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.


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1. The function T|x| = |x-y| is a

|y| |y-x|

linear transformation.

True. It has matrix | 1 -1 |.

|-1 1 |


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

  • for all invertible matrices A.

  • True. A A A-1 A-1 = I.


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  • | 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 |


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  • 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|


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  • 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.


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  • 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.


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  • 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 |


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  • 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 |


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

  • | 0 1 | | 0 1 |

  • numbers k.

  • True:


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  • 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.


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

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


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  • 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.


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

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

  • | 0 1 |


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  • 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.


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  • 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.


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  • 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.


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

  • matrices A representing a reflection.

  • True. For a reflection A2 = I.


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  • 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).


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  • 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 |


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  • 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.


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