midterm1sol.pdf - Math 54 First Midterm Exam, Prof....

Math 54 First Midterm Exam, Prof. Srivastava
September 23, 2016, 4:10pm–5:00pm, 155 Dwinelle Hall.
Name:
SID:
Instructions: Write all answers in the provided space. This exam includes two pages
of scratch paper, which must be submitted but will not be graded. Do not under any
circumstances unstaple the exam. Write your name and SID on every page. Show your work
— numerical answers without justification will be considered suspicious.
Calculators, phones, cheat sheets, textbooks, and your own scratch paper are not allowed.
UC Berkeley Honor Code: As a member of the UC Berkeley community, I act with
honesty, integrity, and respect for others.
Sign here:
Question Points
1 20
2 12
3 10
4 18
Total: 60
Do not turn over this page until your instructor tells you to do so.

Name and SID:
1. (20 points) Circle always true ( T ) or sometimes false ( F ) for each of the following. There
is no need to provide an explanation. Two points each.
For parts (a)-(f), assume A is an arbitrary m × n matrix and R is the reduced row
echelon form of A . For parts (g) and (h) assume A and B are arbitrary n × n square
matrices.
(a) A is invertible if and only if R is invertible. T F
(b) The column space of A is equal to the column space of R .
(c) The null space of A is equal to the null space of R . T
(d) The linear transformations T ( x ) = A x and S ( x ) = R x are the same. T F