T F Every basis of R 3 contains exactly 3 vectors in it

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10/7/2010 FIRST HOURLY Math 21b, Fall 2010 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 Total: 100 1 Problem 1) TF questions (20 points) No justifications are needed. 1) T F The rank of A - 1 is always equal to the rank of A if A is an invertible matrix. Solution: It has to have full rank in order that it is invertible. 2) T F rank( A - B ) = rank( A ) - rank( B ) for all 2 × 2 matrices. Solution: Take A = B = 1 n . 3) T F The row reduced echelon form of an invertible 3 × 3 matrices is invertible. Solution: It is the identity 4) T F The set of cubic polynomials ax 3 + bx 2 + cx + d is a three dimensional vector space. Solution: It is 4-dimensional. 5) T F A system of linear equations has either 0, 1 or many solutions. Solution: This is an important property for systems of linear equations. 6) T F A reflection in the plane at the x axes is similar to the reflection at the y axes. Solution: Just take S ( e 1 ) = e 2 ,S ( e 2 ) = e 1 . This conjugates the two reflections. 2
7) T F Every basis of R 3 contains exactly 3 vectors in it. 9) T F The rank of a 7 × 3 matrix can be 4.
10) T F If { v 1 ,v 2 ,v 3 ,v 4 } is a set of vectors spanning a linear subspace V of R
11) T F If A is a 7 × 5 matrix, then the dimension of ker ( A ) is at least 2.

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