Practice Problems

Practice Problems - Math 110.201: Linear algebra (Spring...

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Math 110.201: Linear algebra (Spring 2008) STUDY SHEET FOR THE FINAL EXAM Scope of the exam Chapters 1–8 (minus § 7.6), plus the additional topics covered in lecture. The final exam is a comprehensive exam covering the content of the entire course. Some additional practice problems In addition to the problems from the midterm review sheets, try the following problems to get additional hands-on practice; any difficulties that you may have with them will help pinpoint gaps in your understand- ing/knowledge. An answer key is not provided because understanding how to check the correctness of your solution is in itself a very good gauge of your grasp of the material. If you are really stuck, ask a friend, or visit your TA or the instructor during office hours. Be sure that you know how to do all previous homework and test problems that you either couldn’t do, or hadn’t done correctly. I also suggest going through some of the true/false problems at the chapter ends. 1. If A is a 3 × 4 matrix such that A 1 2 3 4 = 1 - 1 0 , A 1 1 1 1 = 2 3 - 1 , A 1 0 - 1 - 2 = a b c , then what must the numbers a , b , c be? (Are the numbers even constrained by these equations?) How large could the dimension of the kernel of A be? 2. Find a 3 × 3 matrix A whose 2-eigenspace is spanned by [1 , 1 , 1] > , [1 , 0 , 1] > and whose 3-eigenspace is spanned by [2 , 1 , 0] > . 3. Compute the determinant of the following matrices: A = 0 0 0 1 0 0 2 0 0 3 0 0 1 2 3 4 , ± 0 - A I - I ² , ± A - A I - I ² . 4. Let ~x = [ - 1 , 1] > , ~ y = [2 , 1] > . If possible, find a 2 × 2 matrix A that satisfies the following equalities: ~x · A~x = 6 , ~x · A~ y = 0 , ~ y · A~x = - 6 , ~ y · A~ y = 3 . (Something to think about: Can you compute
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This note was uploaded on 04/01/2010 for the course MATH 110.201 taught by Professor Ha during the Spring '08 term at Johns Hopkins.

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Practice Problems - Math 110.201: Linear algebra (Spring...

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