Dawson College
Mathematics Department
FINAL EXAMINATION
Linear Algebra (Science)
201-NYC-05 sections 4 and 5
December 23, 2020
9:00am to 11:00am
Instructions:
This examination consists of 15 questions.
For this examination, you may use only a Sharp EL-531** calculator and writing
tools (pens, pencils, erasers, ruler).
A webcam that can be used with Zoom must be turned on during the
examination.
Write your solutions on sheets of paper.
When you are finished, take pictures of your solutions. Convert the pictures into a
PDF file.
Submit your PDF file to Moodle. Do not leave the Zoom meeting until receipt of
your submission has been confirmed.
EXAMINATION QUESTIONS
1.
(7 marks)
Find the general solution of the linear system
𝑥1 + 3𝑥2 + 𝑥3 + 2𝑥4 = 1
2𝑥1 + 6𝑥2 + 3𝑥3 + 8𝑥4 = 1
𝑥1 + 3𝑥2 + 2𝑥3 + 7𝑥4 = 0
2.
(7 marks)
For what values of 𝑎 and 𝑏 will the following linear system have
(𝑖) no solution, (𝑖𝑖) one solution, (𝑖𝑖𝑖) infinitely many solutions ?
𝑥+𝑦+ 𝑧= 𝑎
2𝑥 + 3𝑦 + 4𝑧 = 𝑏
3𝑥 + 5𝑦 + 7𝑧 = 𝑎 + 𝑏
3.
(6 marks)
Find a 22 matrix 𝐴 such that
4.
[
1 1
2 1
]𝐴 + [
] = 𝐼
2 3
1 1
(7 marks)
0
Express the matrix [1
1
1 1
0 1] as a product of elementary matrices
1 0
5.
(10 marks)
Let 𝐴 and 𝐵 be square matrices such that 𝐴𝐵 = 0 . Determine whether
each of the following statements is true or false. If true, give a proof. If false,
give a counterexample.
6.
(a)
At least one of the matrices 𝐴 and 𝐵 must be the zero matrix
(b)
If det(𝐴) = 3 , then 𝐵 = 0
(7 marks)
Evaluate the determinant of the matrix
7.
0
2
[
3
4
1
0
3
4
1
2
]
3
0
(5 marks)
Prove that there is no 55 matrix 𝐴 such that
8.
1
2
0
4
𝐴2 + 𝐼 = 0
(6 marks)
Let 𝑢
⃗ and 𝑣 be vectors in 3-space. Find ‖𝑣‖ if
𝑢
⃗  𝑣 = ‖𝑢‖ − 1
‖𝑢
⃗ + 𝑣‖ = ‖𝑢‖ + 1
and
9.
(6 marks)
Find the distance between the point 𝑃0 (1,1,0) and the line passing through
the points 𝐴(0,1,1) and 𝐵(1, −1,2)
10.
(6 marks)
Let  be the plane with parametric equations
𝑥 = 1 + 𝑡1 + 𝑡2 ,
𝑦 = 1 − 𝑡1 + 𝑡2 ,
𝑧 = 2 + 𝑡1 − 𝑡2
Find the point on the plane  that is closest to the point 𝑃0 (1,0, −3)
11.
(6 marks)
Find the general equation of the plane containing the point 𝐴(2,3,4) and the
line 𝐿 ∶ 𝑥 = 1 + 𝑡 , 𝑦 = 2 , 𝑧 = 1 − 𝑡
12.
(6 marks)
Let 𝑊 = {(𝑎 , 𝑏 , 𝑐) | 3𝑎 + 2𝑏 + 𝑐 = 0} . Is 𝑊 a subspace of 𝑅 3 (under
the standard operations) ? If yes, then give a proof. If not, then explain why
not.
13.
(5 marks)
Let 𝑢1 , 𝑢2 , 𝑢3 and 𝑢4 be vectors in a vector space 𝑉 . If the vectors
𝑢1 , 𝑢2 , 𝑢3 and 𝑢4 are linearly independent, prove that the vectors
𝑢1 , 𝑢2 and 𝑢3 are also linearly independent.
14.
(10 marks)
Let
𝑆 = {1 + 2𝑥 , 𝑥 − 𝑥 2 , 𝑥 + 𝑥 2 }
(a)
Show that 𝑆 is a basis for 𝑃2
(b)
Find the coordinate vector of the polynomial 3 − 3𝑥 + 𝑥 2 with respect
to the basis 𝑆
15.
(6 marks)
Find a basis for and the dimension of the following subspace of 𝑀22
𝑊 ={[
3𝑎
𝑎 + 2𝑏
𝑏
] | 𝑎, 𝑏 𝑅 }
−𝑏