solution2 - FIRST HOURLY PRACTICE II Math 21b Fall 10 Name...

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10/7/2010 FIRST HOURLY PRACTICE II 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 are needed. 1) T F Every linear subspace V of R 3 has a unique basis.
2) T F If matrix A is invertible, then rref(A) must be invertible too.
3) T F If A is an invertible matrix, and B = rref(A). Then A 1 = B 1 .
4) T F There is a linear subspace of R 7 that contains exactly seven vectors.
5) T F There exists a 7 × 3 matrix that has rank 7.
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Solution: The circle can not be the kernel. It is not a linear space. 7) T F If a matrix A is similar to a matrix B and A is invertible, then B is invertible.
8) T F A reflection about the line x + y = 1 is a linear transformation.
9) T F If A 2 BA 3 = I 3 for 3 × 3 matrices A, B , then B is invertible. 10) T F For any reflection A about the origin in R
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Solution: The rank of A 2 can be smaller. An example is A = bracketleftBigg 0 1 0 0 bracketrightBigg . 12) T F

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