FinalTestReviewSpring2011

# Give a short explanation for your answer an n n

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Unformatted text preview: r &lt; n (b) r = m and r &lt; n (c) r &lt; m and r = n (d) r = m = n ( pts) Solution: Answers: (a) none or inﬁnitely many; (b) one or inﬁnitely many; (c) none or one; (d) one. Ex. 5. Finish the following sentence. Give a short explanation for your answer. An n × n matrix A is invertible if, and only if, the dimension of its column space C (A) is . . . ( pts) Solution: Answer: “the dimension of C (A) is equal n.” This is the case, since A is invertible when every column of A is its pivot column, that is, when rank r of A is equal n. But we know, that the dimension of C (A) is equal to r. In the test, there will be also a bonus exercise. 2 MATH 441.001 Instr. K. Ciesielski Spring 2011 NAME (print): Review for TEST # 3 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. (a) Find the orthogonal completion V ⊥ of the vector space V spanned by the following three vectors. The description of V ⊥ is understood as: (i) giving (explicitly) a basis for V ⊥ and (ii)...
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## This document was uploaded on 02/19/2014 for the course MATH 441 at WVU.

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