3e-spring2011-chapter_3_tf

3e-spring2011-chapter_3_tf - Chapter 3 3.4.78 Note rst that...

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Chapter 3 3.4. 78 Note first that the diagonal entry s ij of S gives the unit price of good i . If a ij tells us how many dollars’ worth of good i are required to produce one dollar’s worth of good j , then a ij s jj tells us how many dollars’ worth of good i are required to produce one unit of good j , and s 1 ii a ij s jj is the number of units of good i required to produce one unit of good j . Thus b ij = s 1 ii a ij s jj , and B = S 1 AS . True or False Ch 3.TF. 1 T, by Summary 3.3.10. Ch 3.TF. 2 F, by Theorem 3.3.7. Ch 3.TF. 3 T, by Summary 3.3.10. Ch 3.TF. 4 F; The identity matrix is similar only to itself. Ch 3.TF. 5 T; We have the nontrivial relation 3 vectoru + 3 vectorv + 3 vectorw = vector 0. Ch 3.TF. 6 F; The columns could be vectore 1 ,vectore 2 ,vectore 3 ,vectore 4 in R 5 , for example. Ch 3.TF. 7 T, by Theorem 3.3.2. Ch 3.TF. 8 F; The nullity is 6 4 = 2, by Theorem 3.3.7. Ch 3.TF. 9 F; It’s a subspace of R 3 . Ch 3.TF. 10 T; by Definition 3.1.2. Ch 3.TF. 11 T, by Definition 3.2.3. Ch 3.TF. 12 T, by Definition 3.2.1. Ch 3.TF. 13 T; Check that bracketleftbigg 1 0 0 1 bracketrightbigg bracketleftbigg 1 1 1 1 bracketrightbigg = bracketleftbigg 1 1 1 1 bracketrightbigg bracketleftbigg 0 1 1 0 bracketrightbigg . Ch 3.TF. 14 T, by Theorem 3.3.9. Ch 3.TF. 15 T, by Theorem 3.2.8. Ch 3.TF. 16 T, by Summary 3.3.10. Ch 3.TF. 17 F; The number n may exceed 4. Ch 3.TF. 18 T, by Definition 3.2.1 ( V is closed under linear combinations) Ch 3.TF. 19 T, by Theorem 3.4.6, parts b and c.
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