Review
Selected Problems from Old Finals
• Geometry and Vectors
• Systems of Equations
• Linear Transformations
Geometry and Vectors, Short Questions
2015: Points A = [2, 1], B = [5, 7], C = [0, 4] and D form the vertices of a
parallelogram ABCD.
What is the point D? What is the area of the parallelogram ABCD?
2015: Let a = [1, 2, 3], b = [1, 1, 1], and c = [1, −4, 1].
What is the volume of the parallelepiped generated by these vectors?
Find the projection of a onto b + c.
2013: The geometric problem of intersecting two planes P1 and P2 given
respectively by the equations
x1 + 2x2 + 3x3 + 4 = 0 and − 2x1 − 3x2 − 4x3 − 5 = 0
can be written as a matrix equation Ax = b.
Give a possible choice of A, x , and b .
Geometry and Vectors, Short Questions


 
1
1



2
2015: Let L :
+ t 3 be a line in R3 that is parallel to
−1
a
2x + 2y + z = 8. Find a.
2015: Find an equation form of the line [x, y , z] = [3 + 2s, s, 1 − 2s].
2014: What is the shortest distance between points on the parallel lines?
2x − y = 2
4x − 2y = 8
2013: Let d = [−1, k, 2k + 1] where k is a parameter. Find the value of k
for which [−3, −3, 3] is orthogonal to d.
2013: If is possible for the intersection between two planes in R3 with
non-parallel normal vectors to contain two linearly independent vectors?
Lines and Vectors Long Answer: 2015
Line l1 and plane P1 shown in the figure are given in parametric form as
P2
l2
P1
l1
l1 : x = ta
P1 : 
x = q + s1
b1 + s2 b2 where
0
1
a = 1 , q = 1 ,
2
1
 
 
1
0



b1 = 1 , b2 = 2 .
−1
0
Find the intersection point of the line l1 and plane P1 .
Write the parametric form for plane P2 , which contains the line l1 and is
perpendicular to the plane P1 .
Planes P1 and P2 intersect along the line l2 . Write the equation form for
the line l2 .
Geometry and Vectors, Long Question 2013
Let A = (0, 1, 2), B = (1, 0, 3), C = (1, 1, 4). Let ∆ be the triangle whose
vertices are A, B, C .
Find the area of ∆
Prove that ∆ is a right-angled triangle
Find the plane P containing ∆ in equation form, i.e. in the form
ax + by + cz = d with a, b, c, d specified
Find the point D on the plane P such that A, B, C , D form a rectangle.
Systems of Equations, Short Questions
2015: Consider the following linear system written in augmented matrix
form:
5
0 1 2
0 2 1
7
Write the solution or solutions, or state that none exist.
2015: Circle all possible solution sets for linear systems of five equations in
three unknowns.
(a) a unique solution (b) no solutions
(c) an infinite number of solutions (d) exactly two solutions
2014: Circle the one correct answer below. A linear system of three
equations in five unknowns has
(a) always a unique solution (b) either a unique solution or no solutions
(c) either a unique solution or an infinite number of solutions
(d) either no solutions or an infinite number of solutions
(e) either a unique solution or exactly two distinct solutions
Systems of Equations, Short Questions
2014: Consider the system with parameters a and b:
2x
bx
+ y
− 2y
= a
= 6
For what values of a and b (if any) does the system have a unique
solution?
For what values of a and b (if any) does the system have infinitely many
solutions?


1
7
3
p
5. For what p and q does the matrix m have
2013: M = 0
1 pq + 7 8
rank 2?
Choose such p and q. Given an example of a vector b for which the linear
system Mx = b has no solution.
Systems of Equations, Short Questions
2013: Give all solutions to the homogeneous system Ax = 0, where


1 7 7 ··· 7
0 2 7 · · · 7 



7
A = 0 0 3

 .. ..
.. 
.
.
. .
. .
0 0 ···
0
Justify that you have provided all solutions.
13
Systems of Equations, Long Questions
2015: Uno, Duo and Traea are three friends. They all owe money to a
loan shark. All together they owe $600. Duo owes more than Uno. Uno
and Duo combined owe as much as Traea.
 
x1
Let x = x2  be the vector of unknowns, where x1 , x2 and x3 is the
x3
amount of money Uno, Duo and Traea owe, respectively. Describe the
information above as a linear system in the form Ax = b.
Write the system you found above in augmented matrix form.
Solve the system above using Gaussian elimination on the augmented
matrix. How much does each person owe?
Systems of Equations, Long Questions
 
 
 
 
−1
1
1
1







2
4
0
2014: Let v1 =
, v2 =
, v3 =
, b = 2
1
−1
a
b
(a) For what value of a are the three vectors v1 , v2 and v3 listed above
linearly dependent?
(b) For the value of a found in(a), find a linear relation between the three
vectors, that is find scalars α, β, γ such that αv1 + βv2 + γv3 = 0.
(c) For the value of a found in (a), let A = [v1 v2 v3 ] and x = [x1 x2 x3 ]T .
For what value of b does Ax = b have a solution?
(d) For the values of a and b found in (a) and (c), solve Ax = b. Explain
the relation between this solution and the result in (b).
Systems of Equations, Long Questions
2013:
1 3
A=
,
3 8
1 0 −1
B=
−3 1 0
Calculate whenever defined, otherwise state undefined and include a brief
explanation.
(a) A−1
(b) AB
(c) A matrix C such that BC = I2 , where I2 is the 2 × 2 identity matrix.
(d) A matrix D such that DB = I3 , where I3 is the 3 × 3 identity matrix.
Systems of Equations, Long Questions
2013: Let A be a certain 2 × 3 matrix. Suppose that using elementray row
operations the matrix A can be transformed into the matrix
1 0 −7
U=
0 1 4
(a) Find all solutions, if any, of the system Ax = 0
1
(b) Can you find the solution of Ax =
with the given information?
0
Justify your answer.


0
(c) Find all solutions, if any, of the system Ax = A  1 
−2
Linear Transformations, Short Questions
2015: Let A be a 2×2 matrix which represents a reflection across
a line in
2
α
R2 . Suppose that
is an eigenvector with eigenvalue 1 and
is an
3
6
eigenvector with eigenvalue β 6= 1. What are the values of α and β?
2014: Find the projection of the vector [1, −2] only the line through the
origin with direction [3, −4].
2014: Find the matrix that represents reflection across the line through
the origin with direction [3, −4].
2
2013: The linear transformation T : R2 →
R is given
by T(x) = Mx for
2
0
1
1
some matrix M. Given that M
=
and M
=
, find the
1
1
1
0
matrix M.
Linear Transformations, Short Questions
2013: Decide if transformation T - which we define below - is a linear
transformation or not. Justify your answer.
 
x1
3
3

Let T : R → R be a transformation that maps x = x2  into
x3


−x3

1
T (x) = 
x1 + x2 + x3
3
3
2013:
matrix for the
 Find
 the

linear transformation S : R → R given by
x
−y + z
s y  = x + y + z 
z
−x − 2y
Linear Transformations, Long Questions
2014: Let T be a linear transformation. Suppose that
   
   
   
1
−1
0
0
0
0
T 0 =  0  , T 1 = 3 , T 1 = 2
1
−1
0
0
1
2
(a) Calculate T (e1 ), where e1 = [1, 0, 0]T is the unit vector in the
x1 -direction of R3 .
(b) Find the matrix representation of T .
(c) Calculate T 20 (e1 ).
Linear Transformations, Long Questions
2013: Let L be the line 3x = 4y .
(a) Find a unit vector â in the direction of L.
(b) Find the matrix A which represents RefL , the reflection across the line
L.
1
(c) Calculate RefL
0
(d) Find an eigenvector of A associated to the eigenvalue −1.