09-15-10 - Math 54 Discussion Section Problems Mike...

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Unformatted text preview: Math 54 Discussion Section Problems Mike Hartglass September 15, 2010 You should work on the following problems in groups of 3 or 4. Try to get through as many as you can, but you aren’t expected to finish everything. In fact, the answers are largely unimportant; making sure everyone in your group knows how to solve all the problems is what really matters. 1 −2 4 8 . Find a basis for the nullspace of A and the range of A. 1. Consider the matrix A = 2 −4 −3 6 −12 What are the dimensions of these spaces? 1 2 1 03 58 −2 4 1 1 −5 1 2 5 4 2 0 2 0 0 4 1 0 −3 1 1000 0 1 0 0 0 0 0 0 0 0 0 0 π 0 −2 e22 0 −3 0 0 0 0 10 2 12 0 33 0 12 44 1 −17 0 1 2. Find and 3. Find 4. By inspection, determine whether each of the following sets of vectors are linearly independent: (a) 1 2 , 3 −1 , 4 π (b) 3 −6 2 , −4 −1 7 1 0 0 (c) 0 , 0 , 1 −1 0 1 5. Can a 6x9 matrix have a null space of dimension 2? If yes, give an example of such a matrix. If not, explain. Can such a matrix represent an onto linear transformation? What about a one-to-one linear transformation? Either give examples or explain why it is impossible. 6. Determine the relationship between each of the following pairs of determinants (ie, Is one twice the other? Is one the square of the other? etc) (a) (b) (c) a c a c a c b d b d b d and and c a d b 2a 2b cd 2 a c b d and det 7.) If A and B are n × n matrices and AB is invertible, prove that A and B are both invertible. ...
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