253Homework5

# Linear Algebra with Applications (3rd Edition)

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• davidvictor
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ma253 , Fall 2007 — Problem Set 5 This problem set deals mostly with subspaces, spans, linear dependence and independence, and bases. As usual, solve all the problems, then write up and turn in those marked with an asterisk. Given Fall Break, this assignment is due on Friday, October 26 . 1. Problems from the textbook: a. Section 3 . 1 , problems * 18 , * 22 , * 38 , * 48 . b. Section 3 . 2 , problems 1 , * 2 , 3 , * 4 , * 6 , 8 , 10 , 12 , * 14 , * 16 , * 20 , * 32 , * 34 , * 36 , * 37 . c. Section 4 . 1 , problems 1 , * 2 , 3 , * 4 , 5 , * 12 , 13 , 14 . *2. Suppose you have a linearly dependent set { v 1 , v 2 , . . ., v k } . Show that one of the v i can be written as a linear combination of the others. (Yes, we did this in class; write it up carefully.) Is it true that any of the v i can be written this way? *3. Let A be an n × n matrix. Show that the following are equivalent: a. A is invertible. b. A has rank n . c. The transformation T ( v ) = Av is one-to-one. d. The transformation T ( v ) = Av is onto. This is one of the “facts” in the book, so what I’m asking you to do is to convince me that the “fact” really is a fact. The easiest way to approach this

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• Spring '07
• GHITZA

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