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problems-2-223-F07

problems-2-223-F07 - M ATH 223 P ROBLEM S ET 2 D UE 11 S...

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M ATH 223 P ROBLEM S ET 2 D UE : 11 S EPTEMBER 2007 When you hand in this problem set, please indicate on the top of the front page how much time it took you to complete. Reading. § 1.4, § 2.1–2.3. Problems from the book: the starred problems will be graded. 1.4.1, 1.4.3, 1.4.4, 1.4.11abc, 1.4.16, 1.4.26 . 2.1.2, 2.1.3abc, 2.1.6 . 2.2.2abe, 2.2.5 , 2.2.8, 2.2.11. 2.3.2, 2.3.6, 2.3.8, 2.3.11 . Additional problems: 1. Suppose that A is an m × n matrix with m < n . Show that A x = 0 has infinitely many solutions. 2. Let A = bracketleftbigg 1 2 3 1 0 5 bracketrightbigg . a. Is there a 3 × 2 matrix B such that AB = bracketleftbigg 1 0 0 1 bracketrightbigg ? Is there more than one such B ? Can you describe them all? b. Is there a 3 × 2 matrix C such that CA = 1 0 0 0 1 0 0 0 1 ? Is there more than one such C ? Can you describe them all? c. Use Theorem 2.2.1 to prove that only square matrices (might) have inverses. 3. Let f ( n ) = 0 3 + 1 3 + 2 3 + · · · + n 3 = n i = 0 i 3 , for non-negative integers n . Find a,b,c,d,e R such that f ( n ) = a + bn + cn 2 + dn 3 + en 4 , and show using induction that your formula is correct. H INT : You can do this by setting up and solving equations involving
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