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MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 1
ASSIGNMENT 3
Total Marks: 60 (To count 20% of the Final Grade)
Material covered in this Assignment: Modules 6 and 7
Due Date: Thursday, July 9
IMPORTANT!
Please read the discussion forum posts on D2L under
“Messages from your Course Instructor” for details on how to
submit your assignment.
All angles must be expressed in radian measure. Answers
should be given in exact values (no decimals) whenever
appropriate.
Marks will be deducted for not following these instructions.
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 2
1. Which of the following transformations are linear? [9 marks]
a. defined by .
. defined by .
QUESTION #1 CONTINUED ON THE NEXT PAGE
T :R2 → R2 T x
y
!
"
##
$
%
&&=
3x − 5y
y
!
"
#
#
$
%
&
&
T :M22 →M22 T
w x
y z
!
"
##
$
%
&&=
w+ x 1
0 y− z
!
"
##
$
%
&&
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 3
QUESTION #1 CONTINUED
c. defined by .
2. Let T: be the linear transformation defined by
Which, if any, of the following matrices are in ker(T)? Explain your choice(s).
i. ii. iii. iv.
[2 marks]
T :R→ R T (x) = (−3)x
M 22 → P1
T a
c d
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟= a+ d( ) x +
−4 0
6 4
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
0 −2
2 0
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
0 0
0 0
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
3 0
0 3
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 4
3. Define T: by T(X) = AX where A = . [5 marks]
a. Find the kernel of T.
. Give TWO examples of vectors in the kernel.
c. Is T one-to-one? Explain.
R3→ R3
1 1 −1
3 1 −1
−1 −2 2
"
#
$
$
$
%
&
'
'
'
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 5
4. Define T: by T(x, y) = det Find the nullity of the
transformation T and then verify the Rank/Nullity Theorem (Theorem #3).
[3 marks]
5. For what value(s) of k, if any, is the transformation T: defined by
T(x, y) = 2x – k linear? [3 marks]
R2 → R x −4
y −4
"
#
$$
%
&
''
R2 → R
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 6
6. Let T: be a linear transformation defined by
T(x, y, z) = (-z, y + z, -y, 2z – x).
a. Find the standard matrix for T and use it to find the image of (2, 3, -1).
b. Is T an invertible transformation? Explain. [4 marks]
7. Let T be a linear transformation on R2 defined as follows on the standard
asis of R2. T(1, 0) = (6, -1) and T(0, 1) = (-7, 3). Without finding the
standard matrix, calculate T(4, 2). [2 marks]
R3→ R4
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 7
8. Let B = { (1, 1), (3, 5) } and C = { (1, 2), (-1, -1) } be two ordered basis of R2.
a. Find the transition matrix from basis B to basis C.
. Find the transition matrix from basis C to basis B. [5 marks]
c. Show that the transition matrices are inverses of each other.
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 8
9. Calculate the following quantities: [8 marks]
a. Given that z = 2 – i, calculate .
.
c. 2i15 – i36
d.
z2 − z +3
1+ i( ) 3− i( )
1− i( ) −2−3i( )
−1+ 4i
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 9
10. If z is an a
itrary complex number, prove that .
[3 marks]
11. Write the complex number in polar and rectangular form. [2 marks]
12. Write the complex number in polar and exponential form. [3 marks]
Im(z) = − i
2
z − z( )
3ei5π 4
1− 3i
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 10
13. Use the exponential form of a complex number to write in the form
a + ib. [3 marks]
14. Let and . Find AB. [4 marks]
−1+ i( )10
A= 2i 1+ i
−i 2−3i
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ B =
4i −1− i
2+ i 2i
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 11
15. Find the eigenvalues of [4 marks]
A= 1 −i
i 1
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟