ma 351: introduction to linear algebra and its fall 2021
TRANSCRIPT
MA 351: Introduction to Linear Algebra and Its
Applications
Fall 2021, Test Two
Instructor: Yip
• This test booklet has FIVE QUESTIONS, totaling 100 points for the whole test. You
have 75 minutes to do this test. Plan your time well. Read the questions
carefully.
• This test is closed book, closed note, with no electronic device.
• In order to get full credits, you need to give correct and simplified answers and
explain in a comprehensible way how you arrive at them.
Name: (Major: )
Question Score
1.(20 pts)
2.(20 pts)
3.(20 pts)
4.(20 pts)
5.(20 pts)
Total (100 pts)
1
Answer key
1. Fill in blanks: (Show your computations.)
0
B@2
�1
1
1
CA = ?
0
B@1
1
0
1
CA+ ?
0
B@1
0
1
1
CA+ ?
0
B@0
1
1
1
CA
2 + x+ 3x2= ? (1 + x+ x2
) + ? (2� x+ x2) + ? (x� x2
)
3
�1
0
�2
!= ?
2
�1
�1
0
!+ ?
2
�1
0
�1
!+ ?
1
�1
�1
1
!
2
a
I sit lo
1 1 itl
1 of e f eat
You can use this blank page.
3
b 21 329 I txt x 9 2 xxx E x x
2 C 2Ca t OGI C Ca t G3 C t Cz Cz
1 111 1O O 4 4
t.it ii l.iIi it
2 4 3 3 211 txt x2 to E xxx X X
You can use this blank page.
4
e
It l p
0 1 I 2
d Y 89 o 90 1 0 0 2 4 30 1 1 2 0 l l 2
1 1I lit
F sp af fi'D
2. Can you find a linear transformation T from R2to R2
that maps the triangle ABC to
triangle eA eB eC, where
A =
�1
�4
!, B =
1
3
!, C =
�2
4
!,
and
eA =
4
4
!, eB =
�3
�1
!, eC =
�4
�28
!?
(You can assume that T maps A to eA, B to eB, and C to eC.) If so, find the explicit
formula for this linear transformation.
5
1 4 4a 43 4e 4d 4
s 3at be 3Ct 3d 1
2 ga 43 42C t Hd 28
1 E C se1 2 2 14
4 4 4 441 4 70 6 6 18 O o O o
You can use this blank page.
6
Ii tia O
bIsd 3
y s 3 y sty
3. Let A =
1
2
�1
0
!. Find a basis and the dimension for the following subspace of
2⇥ 2 matrices:
M = {B : AB = BA} .
7
B C C 811 fa c b d
ka as
a e atab abt C o
b de a atb d o
2a c 2d La c 22 0
2b c 2bxc o
I off p
2 0 I 2 02 0 I 20
is 4 4 10 2 1 o o O O O O O
You can use this blank page.
8
i iti il
Ac a d pa E B b E
B E ie E aG Plap
aBasis vectors
M Span I 9 f dim m 2
can be chosen as
a
4. Let S = {A2⇥2: A = At} and K = {B2⇥2
: B = �Bt}. (S and K are called the space
of symmetric and skew-symmetric matrices.)
(a) Find a basis and the dimension for S.
(b) Find a basis and the dimension for K.
(c) If you collect the basis vectors for S and K together, do they form a basis for
the space of all 2 ⇥ 2 matrices? If so, write a general 2 ⇥ 2 matrix as a linear
combination of these basis vectors.
9
A et cha a
b c a d freeA 9 a af 4 8 4
s adim S 3 Basis vectors
b 13 61 C sa a a so
b c c bB f if
de de do aBasis vector
dim4 I
You can use this blank page.
10
I c 8 e a1G X G w Caf 8Catch y E YECa Ca Z G YEHence
EY 9 04 1 0189YET 8
Yes the basis vectors for Stk togetherfam a basis for the spaceof 2 2 matrices
They can span and they have 4reetoisintotal Since dim of 2 2 matrices is 4 Hence
theyre automatically linear independent
5. Consider the following linear transformation from R4to R3
:
T (X) =
0
B@1
1
1
1
0
2
2
1
3
�3
1
�7
1
CAX.
(a) Is T onto?
If yes, for a general Y =
0
B@a
b
c
1
CA, find X =
0
BBB@
x
y
z
w
1
CCCAsuch that T (X) = Y .
If no, find an explicit condition on Y such that there is an X such that T (X) = Y .
Find also an example of Y such that there is no X so that T (X) = Y .
(b) Is T one-to-one?
If no, find an Y and X1 6= X2 such that Y = T (X1) and Y = T (X2).
12
I fo i s p0 1 1 4 c a
Zero row
fxkk.getenet is not onto
We need a b za in order for TIX Y to besolvable
eg Y 8 then there is not X s t
THI Y
You can use this blank page.
13
b T is not one to one since there is a
free variable
gSet 8
1 4 iii itO O O O O 9 qX a p
EY WAy 2 48HI P O X 1
Y 1 2 1 w o
2 0
pal Xi 1 y 4 2 0 4 1
Then it I k
fo 71 1I 2 3 7 o
x x
I