253Homework8

Linear Algebra with Applications (3rd Edition)

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ma253 , Fall 2007 — Problem Set 8 Bases, dimension, coordinatization, and the matrix of a linear transformation: all neat stuff. This is due on Wednesday, November 14 . 1. Problems from the textbook: a. Section 3 . 3 , problems 43 , 44 , 47 , * 62 , * 63 , * 64 . b. Section 3 . 4 , problems * 2 , * 4 , * 8 , * 14 , * 18 , * 20 , * 24 , * 30 . c. Section 4 . 2 , problems 63 , * 70 , * 72 , 78 , * 80 . d. Section 4 . 3 , problems 1 , * 3 , * 4 , * 20 , * 22 , * 28 , * 32 . *2. Suppose { v 1 , . . ., v k } is a set of vectors that spans a vector space V , and that T : V - W is a linear transformation that is onto. Show that the set { T ( v 1 ) , . . ., T ( v k ) } spans W . *3. Suppose { v 1 , . . ., v k } is a linearly independent set of vectors in a linear space V , and that T : V - W is a linear transformation that is one-to-one. Show that the set { T ( v 1 ) , . . ., T ( v k ) } is a linearly independent subset of W . *4. Each of the questions on the back consists of a sentence from which one or more words are missing, followed by four possible completions of the sentence. In each case, three of the choices produce sentences that make sense in terms of the concepts we learned thian isomorphisms semester, and the fourth does not. Indicate

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