# Problem 1 TF questions 20 points No justifications...

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10/7/2010 FIRST HOURLY PRACTICE V Math 21b, Fall 10 Name: MWF10 Oliver knill MWF11 Anand Patel Start by writing your name in the above box and check your section in the box to the left. Try to answer each question on the same page as the question is asked. If needed, use the back or the next empty page for work. If you need additional paper, write your name on it. Do not detach pages from this exam packet or un- staple the packet. Please write neatly and except for problems 1-3, give details. Answers which are illegible for the grader can not be given credit. No notes, books, calculators, computers, or other electronic aids can be allowed. You have 90 minutes time to complete your work. 1 20 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total: 110 1
Problem 1) TF questions (20 points) No justifications needed 1) T F The plane x + y + z 1 = 0 is the kernel of a linear transformation T .
2) T F For any matrix A the identity im ( A 2 ) = im (rref ( A 2 )) holds.
3) T F For any matrix A , one has ker ( A ) = ker (rref ( A )).
4) T F There is a 4 × 8 matrix whose kernel is 3-dimensional.
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Solution: Look for rotations. I 2 is a rotation by 180 degrees. So, a rotation by 90 degrees will do. This is by the way a matrix representation of the complex number i . If you ever wondered whether i , the square root of 1 exists, here it is. 6) T F If three vectors v 1 , v 2 , and v 3 are in a plane, they are independent.