MAT 242 Study Guide - Midterm Guide: Ti-89 Series, Row And Column Spaces, Proctor
C. HECKMAN
Test 2 A 242
Name:
Instructions:
•The exam consists of six (6) problems, some of which may have several parts. It has five (5) pages
(including this one); you should make sure that you have all of them before you start.
•Turn off your cell phone or any communications device (if you have one) and put it away, and remove
any headphones before beginning the test.
•Show all work in detail or your answer will not receive ANY credit. Write neatly and box all answers.
If you need extra space for work, you may get scratch paper from a proctor; do not use your own paper.
•Make sure you read the problems and answer everything that is asked. If you are asked to use a partic-
ular method, you must use that method to receive full credit. If you are not told to use any particular
method, you may use any method mentioned in class.
•No calculators with Qwerty keyboards or ones like the Casio FX-2, TI-89, or TI-92 that do symbolic al-
gebra may be used. If you use your calculator for a calculation, make sure you indicate which expression
you are entering into your calculator; do NOT just give a final answer.
Honor Statement: By signing below I confirm that I have neither given nor received any unauthorized
assistance on this exam. This includes any use of a graphing calculator beyond those uses specifically autho-
rized by the School of Mathematical and Statistical Sciences and my instructor. Furthermore, I agree not
to discuss this exam with anyone until the exam testing period is over. In addition, my calculator’s memory
and menus may be checked at any time and cleared by any proctor or School of Mathematical and Statistical
Sciences instructor.
Signature
1. [30 25 points] For the matrix Abelow, find a basis for the null space of A, a basis for the row space of
A, a basis for the column space of A, the rank of A, and the nullity of A.
A=
5−25 −3 2
−5 25 0 −4
2−10 −3 2
−3 15 2 −2
Solution: The reduced row echelon form is R=
1−5 0 0
0 0 1 0
0 0 0 1
0 0 0 0
.
To find a basis for the null space, you need to solve the system of linear equations A~x =~
0,
or equivalently, R~x =~
0. Parameterizing the solutions to this equation, and writing the answer in
expanded form, produces
x1
x2
x3
x4
=a·
5
1
0
0
so
5
1
0
0
is a basis for the null space of A. The nullity is the number of vectors in this basis, namely 1.
A basis for the row space can be found by taking the nonzero rows of R:
{[ 0,0,0,1 ] ,[ 0,0,1,0 ] ,[ 1,−5,0,0 ]}
A basis for the column space can be found by taking the columns of Awhich have pivots in them, so
5
−5
2
−3
,
−3
0
−3
2
,
2
−4
2
−2
is a basis for the column space of A.
Lastly, the rank of Ais the number of vectors in a basis for the row space (or column space)
of A, so the rank of Ais 3.
Grading: +5 points for finding a basis for the null space, +5 points for each of: a basis for the
row space, a basis for the column space, the nullity, and the rank. Grading for common mistakes:
−3points for forgetting a variable in the parameterization; −3points for choosing columns of R
for the column space of A;−3points for choosing rows from Afor the row space of A;−7points
for choosing the non-pivot columns of Afor the null space of A.
2
Document Summary
Instructions: the exam consists of six (6) problems, some of which may have several parts. If you need extra space for work, you may get scratch paper from a proctor; do not use your own paper: make sure you read the problems and answer everything that is asked. If you are asked to use a partic- ular method, you must use that method to receive full credit. If you use your calculator for a calculation, make sure you indicate which expression you are entering into your calculator; do not just give a nal answer. Honor statement: by signing below i con rm that i have neither given nor received any unauthorized assistance on this exam. This includes any use of a graphing calculator beyond those uses speci cally autho- rized by the school of mathematical and statistical sciences and my instructor. Furthermore, i agree not to discuss this exam with anyone until the exam testing period is over.
