Linear Algebra with Applications (3rd Edition)

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ma253 , Fall 2007 — Problem Set 7 Most of this assignment deals with bases and dimension, both in R n and in more general vector spaces. Linear transformations also come in, and the rank and nullity theorem will play role. This is due on Wednesday, November 7 . 1. Problems from the textbook: a. Section 3 . 3 , problems * 18 , * 22 , * 26 , * 28 , * 30 , * 36 , * 38 . b. Section 4 . 1 , problems * 18 , * 26 . c. Section 4 . 2 , problems * 22 , * 24 , 25 , * 27 , * 32 , 33 , * 34 , * 42 , * 50 , * 54 , * 60 . *2. In problem 34 from section 4 . 2 , you are asked to work in the space V of all infinite sequences of real numbers and to consider what is usually called the right shift operator , defined by R ( x 0 , x 1 , x 2 , x 3 , . . . ) = ( 0, x 0 , x 1 , x 2 , x 3 , . . . ) . Show that ker ( R ) = { 0 } , where 0 means the zero vector in V , i.e., the se- quence ( 0, 0, 0, 0, . . . ) . Is R invertible? *3. For linear transformations T : R n R n (or, equivalently, for n × n matrices), we showed that if ker ( T ) = { 0 } then T is invertible. Compare this with the previous problem. What’s going on? *4. We can also define the left shift operator L on the space of all infinite sequences: L ( x 0 , x 1 , x 2 , x 3 , . . . ) = (
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