Download Question Sheet 1 1. Let u = (−1,1,2) v = (2,0,3) w = (1,3,12

Question Sheet 1
1. Let
u = (−1, 1, 2)
v = (2, 0, 3)
w = (1, 3, 12)
be vectors in R3 .
(a) Compute 13 (3u − v + 4w).
(b) Find the vector z such that 3(2u + z) + w = 2z − v.
(c) Find the scalar λ such that u + λv is parallel to w. For this value of λ, decide if u + λv
has the same (or opposite) direction as w.
(d) Compute the dot products u · w and v · w. Hence find the scalar µ such that u + µv
is perpendicular to w.
2. (a) Let P and Q denote points in R3 with position vectors p and q respectively. Find the
position vector a of a point A on the line through P and Q which is 3 times as far from
Q as from P : does this fix the point A uniquely?
(b) Consider a triangle ABC. Let M be a point on AB such that BM = 2M A and N a
point on AC such that CN = 2N A. Using vectors, prove that the line segment M N is
parallel to BC and M N = 31 BC.
3. Find a formula for the angles θ1 , θ2 and θ3 made by a position vector a = (a1 , a2 , a3 ) with
the three coordinate axes. Evaluate
cos2 θ1 + cos2 θ2 + cos2 θ3
4. (a) Find the parametric equation of the line in R3 passing through the point (4, 1, 5) and
parallel to the position vector of (1, 0, 1).
(b) Find the parametric equation of the line in R3 passing through the points (2, −7, 12)
and (2, 9, −6).
(c) Show that the lines in parts (a) and (b) intersect, and find their point of intersection.
5.
Find a parametric equation of a line in R4 passing through the points P = (0, 1, −2, 2)
and Q = (1, 1, −1, 2). What is the angle between the vectors v = (0, 1, −1, 2) and w =
(1, 1, −1, 7) in R4 .
Question Sheet 2
6. (a) Find the Cartesian equation of the plane Π1 in R3 passing through the point (2, 1, 3)
and perpendicular to the vector n = (1, −1, 0).
(b) Find the Cartesian equation for the plane Π2 passing through the points (0, 0, 0),
(1, 0, −2) and (0, 1, 1).
(c) Find the angle between the planes Π1 and Π2 . (The angle between two planes is the
angle between their normal vectors).
(d) Does either the origin O or the point P = (3, 2, 2) lie on Π1 ?
7. Two planes in R3 are either parallel or intersect in a line.
(a) Describe briefly how you would decide whether or not two planes are parallel.
(b) How would you find the line of intersection of two non-parallel planes?
Find the line of intersection of the two planes Π1 and Π2 of Question 2.
8. Decide if the line through the points
P = (2, 2, 0)
and
Q = (5, −1, 3)
and the plane
4x + y − 3z = 10
(a) intersect in a single point (if so, find it), or
(b) the line is parallel to the plane and not in it; or
(c) the line is contained in the plane.
Justify your answer.
9. Show that, for any two vectors u and v in Rn ,
ku + vk ≤ kuk + kvk
(triangle inequality)
[Hint: Begin by squaring both sides and use the dot product: ku + vk2 = (u + v) · (u + v).]
Question Sheet 3
10. (a) Express each of the vectors w, 0 and q given below as a linear combination of u and
v where possible:
w = (3, −2, 4) 0 = (0, 0, 0) q = (7, −5, −7)
u = (2, −1, 1) v = (−1, 1, 3)
(b) Decide which of the following sets are a basis for R3 . Give reasons for your answers.
Which sets are linearly independent? Which sets span R3 ?
S1 = {u, v, w, q}
S2 = {u, v}
S3 = {u, v, w}
S4 = {u, v, q}
(c) Let v1 , v2 , and v3 be three vectors in R3 . For simplicity assume that they are not all
parallel to each other. We say these vectors are coplanar, if there is a plane through the
origin in R3 that contains all of them. Show that v1 , v2 , v3 are coplanar if and only if
they are linearly dependent.
(d) Find a value of r for which the position vectors u, v and r = (0, 5, r) are coplanar.
11.
(a) Show that the set of three vectors {(1, 1, 1), (1, −1, 1), (0, 0, 1)} is linearly independent,
hence a basis for R3 .
(b) Show that the set of vectors {(1, 1, 0, 0), (1, 0, 1, 0), (0, 0, 1, 1), (0, 1, 0, 1)} ⊂ R4
linearly dependent, hence not a basis.
is
[Simplifying hint: One pleasant way to do this is the following: first see that the two first
vectors are linearly independent, then convince yourselves that, even more, the first three
vectors are also linearly independent. This means that if there is to be a linear relation
amongst these four vectors, it must be introduced by the fourth one. In other words,
the fourth one should be expressible as a linear combination of the rest. Find that linear
combination! Or just do it the routine way following definitions]
Is the vector p = (1, −1, 4, 2) in the linear span of these four vectors ?
12. Find a basis for these subspaces of R3 : (a) The plane x − 2y + z = 0
(b) The xy plane.
Find a basis for the hyperplane in R4 given by : (c) 2x1 − 2x2 + x3 + 3x4 = 0. (Hint: use
the same method as in (a) ).
Question Sheet 4
13. Each of the following systems of equations gives the intersection of three planes. In each
case find the solutions by Gaussian reduction and describe the pattern of intersecting
planes which it represents.
(1)


 x+y+z =2
2y + z = 0


−x + y − z = −4


x + y + z = 2
2y + z = 0


−x + y = −4
(2)
(3)


x + y + z = 2
2y + z = 0


−x + y = −2
14. Is it possible for a system of two equations in three variables to have just one solution?
Explain your answer:- if you think the answer is yes, find an example of such a system; if
not, give a geometric reason why not.
15. Which of the following matrices are in reduced echelon form? Explain your answers.

1
0
0

0
0
1
1
0
2
0
3
1
1
0
2
0
1 0
1 0
 
0
1
 
0
0
16. Solve the following system of equations using Gaussian elimination.



x1 − x2 + x3 + x4 + 2x5 = 4


x1 − x2 + 2x3 + 3x4 + 4x5 = 7
−x1 + x2
+ x4 − x5 = −3
Show that your solution is of the form x = p + su + tv where x is the vector whose
components are the unknowns. p is a solution for the above system, and u and v are
solutions of the following system



x1 − x2 + x3 + x4 + 2x5 = 0


x1 − x2 + 2x3 + 3x4 + 4x5 = 0
−x1 + x2
+ x4 − x5 = 0
(Note and suggestion: the second system has the same coefficients as the first system, but
its constants are all zeros. The second system is called the homogenous linear system of
equations associated to the first one. Solve the two systems side-by-side to see the idea
more clearly).
Finally, is it possible to replace the column of constants in the above system of linear
equations to obtain a system that is inconsistent?
Question Sheet 5
17. Consider the system of equations for the unknowns x, y and z, where µ is a parameter.
x + 2y − z = 2
x−y+z =5
3x + 3y − z = µ
For what values of µ does this system have
(a) no solutions,
(b) exactly one solution,
(c) infinitely many solutions?
In each case, find the solutions (if any).
18. Use the method of row operations to find the inverse of each of the following matrices (if
the inverses exist).

1

(a) A = 2
0
2
3
1

3
0
2

1

(b) B = 2
0

2 3
3 0
1 6
19. Write the system of linear equations

x1 − x2 + 5x3 + 2x4 = 1





x1 + 7x3 + 5x4 = −2

x2 + 3x3 + 6x4 = 0




x1 − x2 + 5x3 + x4 = 1
in matrix form (Ax = b). Use the inversion algorithm to find the inverse of the coefficient
matrix (A), and use A−1 to find the solution of the system.

20.
3

A=
1
−1
1 0
2 −1
1 0

5
3
1

1
 2 
p=

−1
1

(a) Find the vector b such that Ap = b.
(b) Express b as a linear combination of the column vectors of A. (Hint: you can do this
just by looking carefully at part (a). Don’t try to solve any system of equations yet!)
(c) Is this linear combination unique? Find the general solution of the ‘homogeneous’
equation Ax = 0 and use it to write down the general solution of Ax = b (in other
words, invoke the “Principle of Linearity”). Use that to find another linear combination
of the column vectors of A which equals b.
Question Sheet 6
21. Evaluate the following determinants using
row or column.
5
7
2 3 1
0
2
(b) (a) 1 0 2 11
2
7 1 1
23 57
the cofactor expansion along an appropriate
2
0
0
1
3 0 0 −1
3 t −2 (c) −1 5 3 2 1 1 For what values of t is the determinant in (c) equal to zero?
22. (a) Find the inverse matrix, using the cofactor (adjugate) method, or show that none
exists:






2 0 −3
1 0 2
1 2 0
A =  0 3 1 ;
B = 2 1 3;
C = 0 1 1 
−1 4 2
0 −1 1
2 1 −1
(b) Solve the system of equations Cx = b for b = ( 5
4 2)
T
using Cramer’s rule.
23. Evaluate the following determinants using row operations to simplify the calculation.
(Remark: Starting with a square matrix A, apply Gaussian elimination to bring A to a
upper-triangular form B (this is the first stage of the Gaussian elimination, before one
sets out to make the entries above leading ones zero). Now, knowing |B| work your way
backwards to find |A|. Keeping in mind that we know what each row operation does to
the determinant. However, it may not be necessary to follow this argument to the end (or
at all!), as sometimes in the midst of this algorithm the matrix finds a simple enough form
for determinant calculation using other properties of determinants we have learned.)
(a)
1
2
1
2
−4
−7
2
−10
24. A is 3 × 3, det(A) = 5.
det(A−1 AT ).
3
5
6
14
Find
2
1
0
4
(b)
det(3A),
1
1
2
2
−1
4 −1
7 4
8 −2
0 5
9 0
det(A2 ),
3
3
6
5
9
0
8
0
7
2
det(2A−1 ),
det (2A)−1 ,
25. Consider the vectors
a = (2, 1, 3)
b = (2, −1, 4)
c = (−1, 1, 0)
(a) Find a × b and b × c.
(b) Using cross products, find a Cartesian equation for the plane which passes through the
points (2, 1, 3), (3, 1, 1) and (2, 2, 4).
Question Sheet 7
26. Show that each of these sets is a basis of R3 :
B = {v1 , v2 , v3 } = {(1, 0, 1), (1, 1, 3), (0, 0, 1)}
and
b = {b
b2 , v
b3 } = {(1, 1, 1), (1, −1, 0), (0, 1, −1)}
B
v1 , v
(a) Write down the transition matrix P from B-basis to standard basis.
b
Write down the transition matrix Q from B-basis
to standard basis.
(b) If v = (3, −1, 2), find v in the B basis
(c) Given that x = 2b
v1 + 1b
v2 + 3b
v3
b basis, find x in the B basis
in the B
b basis to B-basis.
(d) Write down the transition matrix from B
b to standard basis and then from
(Note: You may want to change coordinates from B
standard basis to to B-basis).
27.
M = {e1 , e2 } = {(2, 1), (−1, 1)}
(a) Show that M is a basis of R2 .
standard basis.
v = (−1, 2)
T (x, y) = (7x − 2y, −x + 8y)
Write down the transition matrix from M -basis to
(b) Find [v]M , ie find v in the M -basis.
(c) Write down the matrix of the linear transformation T , T : R2 → R2 with respect to
the standard basis. Call it A.
(d) Find the matrix of T in M -basis. Call it B.
Describe geometrically the transformation T as a map from R2 → R2 , in terms of M -basis.
(e) Find the standard basis, and the M -basis of the vector T (v), without using any transition matrices (and by using parts (a) and (d)).
(f) Verify your answer to part (e) using the transition matrix P from M -basis to standard
basis.
28. Let T : R2 → R2 be the linear map in the previous question. Let
u1 = (4, 1)
u2 = (2, 3).
(a) Find the area of the parallelogram F formed by u1 , u2 .
(b) Convince yourself that the image of F under T , denoted T (F ), is the parallelogram
formed by T (u1 ), T (u2 ). Then calculate the area of T (F ).
(c) Find det(A).
(d) Verify that Area(T (F ))/Area(F )=det(A).
This is one conceptual description of determinants. It is true, in general, that if A is the
matrix of a linear map T : R2 → R2 , then T will take any geometric figure in the plane
to another one, and the area of the image will be det(A) times the area of the original
figure. A similar statement is true in higher dimensions: for example in R3 ; but one should
replace “area” by “volume ” in this case.
One easy case is when the matrix is a 3 by 3 scalar, say 5I3 . Then T will stretch any solid
in R3 by a factor of 5 in each direction. Intuitively, it is easy to see that this transformation
multiplies the volume by 5 × 5 × 5. But that number is exactly the determinant of 5I3 .
Question Sheet 8
29. Let
A=
1
3
4
.
2
(a) Find the eigenvalues and the corresponding eigenvectors for A.
(b) Find an invertible matrix P such that P−1 AP is diagonal.
(c) Calculate P−1 AP to check your answer.
(d) Find A5 .

30. Let
3
B = 5
1

−1 2
−3 5 .
−1 2
(a) Show that λ = 1 is an eigenvalue of B and find the corresponding eigenvector(s).
(b) Find the characteristic equation of B. Find all the eigenvalues. (Hint: Use the fact
that λ = 1 is one of the eigenvalues to factor the characteristic polynomial).
(c) Find the corresponding eigenvectors for B.
(d) Show that B is diagonalizable. Hence, find an invertible matrix P such that P−1 BP =
D is diagonal. Check your answer for P. (Hint: it is easier to show that P is invertible
and BP = PD to avoid having to calculate P−1 ).
(e) How would you describe the linear transformation T (x) = Bx?
31. For each of the following matrices decide which are diagonalizable, either over R or C.
A=
3
1
−6 −5
2
B=
C=
10
8
3
5
1
2 −1
D=
E=
−9 −1
−4 2


1 1 1
F =  0 1 −1 
1 0 2
2
1
−3
2
(You are NOT being asked to find the change of basis matrices P for any of these.)
32. Show that 0 is an eigenvalue of a matrix A if and only if A is not invertible.
Question Sheet 9
33. Decide which of the equations below are linear and which are not.
Find the general solution of each linear equation.
(a)
dy
− 2y = e3x
dx
(b)
x
(c)
xy
(d) 2x
(e)
dy
− 4y = x + 1
dx
dy
− y 2 = 3x2 e2y/x
dx
√
dy
=2−2 x−y
dx
2
e−(3x
+5) dy
dx
= xy 2
34. Show that the following sets of functions are linearly independent
{eλx , xeλx }
{1, sin x, cos x}
(λ a real constant)
Is the set {1, sin2 x, cos2 x} linearly independent? Justify your answer.
35. Find the general solution of each of the following differential equations:
(a)
dy
d2 y
−4
+ 3y = e2x
2
dx
dx
(b)
d2 y
+ y = x2 + x
2
dx
(c)
d2 y
dy
+
− 2y = ex
2
dx
dx
(d)
d2 y
dy
+2
+ y = 2x
2
dx
dx
Find the particular solution of (d) which satisfies y(0) = 0,
y 0 (0) = 1.
36. Find the general solution of the system of differential equations

dy1


= 4y1 − 2y2 ,
dx

 dy2 = y + y .
1
2
dx
Find also the solution which satisfies the initial conditions y1 = 0, y2 = 3 when x = 0.
37. Find the general solution of the equation
dy
+ y cot x = 1
dx
Then find the solution which satisfies the condition y( π2 ) = 1
38. Find the solution of the initial value problem
dy
d2 y
−
− 2y = 10 sin x
dx2
dx
y = −3,
dy
π
= −1 when x = .
dx
2