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MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 1
ASSIGNMENT 2
Total Marks: 85 (To count 25% of the Final Grade)
Material covered in this Assignment: Modules 3, 4, and 5
Due Date: Thursday, June 18
IMPORTANT!
Please read the discussion forum posts on D2L under
“Messages from your Course Instructor” for details on how to
submit your assignment.
Marks will be deducted for not following these instructions.
It is expected that techniques from this class (specifically
solving systems with matrices) will be used for solving
problems when applicable.
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 2
1. Let . Find det(A). [4 marks]
A=
0 −2 1 4
3 3 2 −1
2 6 5 7
0 0 −4 0
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 3
2. Let . Find the adjoint of A. [4 marks]
A=
3 −1 1
0 2 4
4 −2 5
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 4
3. Use Cramer’s rule to solve . [5 marks]
−2x1 −5x2 + 4x3 = 21
−5x1 −5x2 + x3 = 21
−4x2 − 4x3 = 8
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 5
4. Suppose A, B, and C are 3 3 matrices for which det(A) = -2, det(B) = 4, and
det(C) = 3. Find the value of the following determinants.
a. det(C-1ATB2B-1), assuming B-1 and C-1 exist.
. det(2A-1Ct), assuming A-1 exists. [3 marks]
c. det(4B-1C2), assuming B-1 exists.
5. Suppose and = det(A) = 3. What are the values of the
following determinants? [3 marks]
a.
.
c.
€
×
€
A =
0 5 8 2
a b c d
− XXXXXXXXXX
e f g h
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(
(
(
(
€
A
€
5 0 8 2
a c d
7 −3 10 6
f e g h
€
XXXXXXXXXX
a + e b + f c + g d + h
− XXXXXXXXXX
e f g h
€
XXXXXXXXXX
a + 2b b c d
XXXXXXXXXX
e + 2 f f g h
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 6
6. Let . For what values of t does A-1 exist? [4 marks]
€
A =
2 t 6
0 2 t
1 0 2
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#
$
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' '
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 7
7. Let . Suppose addition and scalar multiplication are defined
using the following non-standard rules.
[5 marks]
where c is any real number.
a. Find the result of (1, XXXXXXXXXX, -3) under the above operations.
. Find the result of 3(5, -1) under the above operations.
c. Show that V, with respect to these operations of addition and scalar
multiplication, is not a vector space by showing that one of the vector
space axioms does not hold.
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V = (x,y) x,y ∈R{ }
€
x1,y1( ) + x2,y2( ) = y1 + y2,x1 + x2( )
€
c x1,y1( ) = cx1,2cy2( )
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 8
8. Let . Define addition on V as follows:
[4 marks]
a. Prove addition axiom #3 (Addition is commutative).
. Find the zero vector.
9. a. Is the set of vectors a subspace of R3?
[5 marks]
QUESTION #9 CONTINUED ON THE NEXT PAGE
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V = (x,y) x,y ∈R{ }
x1, y1( )+ x2 , y2( ) = x1 + x2 − 2, y1 + y2 −1( )
€
W = (x1,x2,x3)∈R
3 x2 = x1 + x3{ }
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 9
QUESTION #9 CONTINUED
b. Is the set of M of all matrices of the form a subspace of M2x2?
10. For what value(s) of r (if any) is the vector (r, 20, -6) a linear combination of
the vectors (3, 6, 9) and (4, -2, 3). [4 marks]
a a+3
0
!
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#
$
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MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 10
11. Is the set of vectors { (2, -1, 5), (1, 0, 7), (-6, 3, -15)} linearly independent or
linearly dependent? Explain your conclusion. [2 marks]
12. Is the set of polynomials {3x2, x2 - 10x + 15, 10x -15} a basis for P2? Refer to
the definition of a basis for full marks. [4 marks]
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 11
13. a) Use the Gram Schmidt Orthogonalization Procedure to transform the basis
{ (1, 1, 0), (1, 0, 1), (1, 1, 1) } into an orthogonal basis for R3. [5 marks]
b) Use the dot product to verify your result from part (a).
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 12
14. Find a basis for the column space of the following matrix. Give the basis in
educed echelon form. What is its rank?
. [4 marks]
1 −3 4 −2 5 4
2 −6 9 −1 8 2
2 −6 9 −1 9 7
−1 3 −4 2 −5 −4
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#
$
$
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MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 13
15. Which of the following vector(s) is/are unit vectors? Circle the co
ect
esponse and justify your work in the space provided. [3 marks]
a. (1, 1, 1) b. c. d.
16. Find the equation of a line through the point (2, -3, 1) in a direction orthogonal
to the line . Give your answer in both parametric and
symmetric form. [4 marks]
1
3
, 1
3
, 1
3
⎛
⎝
⎜
⎞
⎠
⎟
1
3
,1
3
,1
3
⎛
⎝
⎜
⎞
⎠
⎟
1
6
2,−1,1( )
x +1
3
=
y−1
2
=
z+ 2
5
MATH 1057EL XXXXXXXXXXSPRING 2020 ASSIGNMENT FILE Page 14
17. Find the equation of a plane that passes through the points (15, 5, 2), (6, 2, 1)
and (10, 3, 2). Does the point (-2, -5, -3) lie on the plane? [5 marks]
18. Let .