Friday November 20
START: 16:10
DURATION: 110 mins
University of Toronto
Department of Mathematics
MIDTERM EXAMINATION II
SOLUTIONS
MAT223H1F
Linear Algebra I
EXAMINERS: T. Bazett, I. Biborski, N. Garcia-Fritz, S. Homayouni-Boroojeni, A. Kolpakov, H. Nuchi, S. Uppal
Last Name (PRINT):
Given Name(s) (PRINT):
Student NUMBER:
Student SIGNATURE:
Tutorial Group:
Instructions.
1. There are
63
possible marks to be earned in this exam. The examination booklet contains a total of 11 pages. It is your
responsibility to ensure that
no pages are missing from your examination
. Do not detach any pages from your examination.
2. You may write in black ink, blue ink, or in pencil. Do not write in red ink, and ensure that your solutions are LEGIBLE.
3. No aids of any kind are permitted. CALCULATORS AND OTHER ELECTRONIC DEVICES (INCLUDING PHONES)
ARE NOT PERMITTED.
4. Have your student card ready for inspection.
5. There are no part marks for Multiple Choice (MC) questions.
6. For the full answer questions, write the solutions on the question pages themselves. You may use the two blank pages at
the end for rough work.
The last two pages of the examination WILL NOT BE MARKED unless you
clearly
indicate
otherwise on the question pages.
7. For the full answer questions, show all of your work and justify your answers
but do not include extraneous information.
8. Only solutions that are written in ink with no correction fluid or correction tape can be considered for re-grading.
FOR MARKER USE ONLY:
MC
Q1
Q2
Q3
Q4
Q5
Q6
TOTAL
1

Part I - Multiple Choice Answer Key
(1): (A)
(2): (B)
(3): (A)
(4): (C)
(5): (A)
1. Which of the following statements are TRUE?
(i) It is possible for a linear transformation from
R
2
to
R
2
to transform a parallelogram to a circle.
(ii) It is possible for a linear transformation from
R
2
to
R
2
to transform a parallelogram to a line segment.
(iii) It is possible for a linear transformation from
R
2
to
R
2
to transform a parallelogram to a triangle.
(A)(ii) only(B)(i) only(C)(i) and (iii) only(D)(i) and (ii) only(E)(i), (ii), and (iii)Answer:(A) (ii) only.
(i)
:
False. Linear transformations take straight lines to straight lines (or to a single point in certain situations). Since the sides
of a parallelogram are straight lines, it follows that linear transformations cannot take a parallelogram to anything other than a
polygon (which a circle is not).
(ii)
:
True. As before, linear transformations take straight lines to straight lines. However, it is possible for a linear transformation
to take the two adjacent sides of a parallelogram, and map them onto the same line.
For example, suppose
T
:
R
2
→
R
2
is given by
T
(
x
) =
1
0
1
0
x
. What does
T
do to the square
S
with vertices (0
,
0)
,
(1
,
0)
,
(0
,
1)
and (1
,
1)?
We find that
T
0
0
=
0
0
,
T
1
0
=
1
1
,
T
0
1
=
0
0
, and
T
1
1
=
1
1
. We see that
T
takes the
S
to the line
segment from (0
,
0) to (1
,
1).
1
1
1
1
T
(iii)
:
False. Linear transformations take parallelograms to either parallelograms, straight lines, or single points. They cannot
take a parallelogram to a triangle. Suppose we have a parallelogram P:
A
B
C
D
Since
--→
AB
=
--→
CD
, we know that
T
takes the line segments
AB
and
CD
to parallel line segments of equal length. The same is