FOR MORE CLASSES VISIT www.tutorialoutlet.com This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority of the Academic Board.

1. Consider the points A = (2, −1, −1), B = (−1, 2, −1) and C =
(−2, 1, −3) in R3 .What is the distance between the points
B and C?
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This paper is part of an examination of the College counting towards
the award of a degree.
Examinations are governed by the College Regulations under the
authority of the Academic Board.
PLACE this paper and any answer booklets in the EXAM
ENVELOPE provided Candidate No: . . . . . . . . . . . . . . . . . . . . . . . . . .
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Examination
4CCM113a Linear Methods
Summer 2017
Time Allowed: 2 hours
This paper consists of two sections, Section A and Section B. Section

2. A
contributes 75% and section B contributes 25% of the total marks for
the
paper. Answer all questions. NO calculators are permitted. DO NOT
REMOVE THIS PAPER
FROM THE EXAMINATION ROOM
TURN OVER WHEN INSTRUCTED
c
2017 King’s
College London 4CCM113A
SECTION A
In section A each question carries 5.0 marks. Note that for any wrong

3. answer 1.0 will be deducted from your total. There is only one correct
answer for each question. Put all your answers in the grid. A 1. Let R3
be equipped with the standard dot or inner product. Consider the
points
A = (2, −1, −1), B = (−1, 2, −1) and C = (−2, 1, −3) in R3 .
What is the distance between the points B and C?
(A) A 2. 3 (B) √ 6 (C) √ 17 (D) √ 15 (E) none of these Let R3 be
equipped with the standard dot or inner product. Consider the
points A = (2, −1, −1), B = (−1, 2, −1) and C = (−2, 1, −3) in R3 .
What is
−→
−→
the cosine of the angle between the directed line segments AC and
AB?
(A) A 3. √ √3

4. 12 (B) √1
4 (C) √5
17 (D) √1
12 (E) none of these Let R3 be equipped with the standard dot or
inner product. Consider the
points A = (2, −1, −1), B = (−1, 2, −1) and C = (−2, 1, −3) in R3 .
What is a
−→
parametric equation of a line parallel to AC that passes through B?
(A) r = t(−4, 2, −2) + (−1, −1, −1)
(C) r = t(−4, 2, −2) + (−1, 2, −1)
(E) none of these -2- (B) r = t(1, 2, −2) + (−1, 2, −1)

5. (D) r = t(1, 2, −5) + (−1, 2, −1) See Next Page 4CCM113A
A 4. Let R3 be equipped with the standard dot or inner product.
Consider the points
A = (2, −1, −1), B = (−1, 2, −1) and C = (−2, 1, −3) in R3 . Give a
coordinate
equation of the plane that passes through A, B and C.
(A) −x − y + z = −2
(D) −x − y + 2z = −7 (B) −x − y + 2z = −2
(E) none of these (C)−2x − y + 3z = −2 A 5.
Let R3 be equipped with the standard or dot inner product and i, j and
k
be the standard basis of R3 . What is the exterior product u × v of the
vectors

6. u = 2i − j + 3k and v = i − 4j − 2k?
(A) 4i + 7j − 12k
(D) 14i + 7j − 7k (B) i + 7j + 7k
(E) none of these (C)14i + 5j − 3k A 6. Let R3 be equipped with the
standard or dot inner product and i, j and k be
the standard basis of R3 . What is the area of the parallelogram
defined by the
vectors u√= i − j + k and
− 2k ?
√ v = 4i − 3j √
√
(A) 62

7. (B) 11
(C) 3
(D) 3 5
(E) none of these A 7. Which of the following choices of vectors is a
basis for the linear subspace of R3
defined by the equation 2x − y − 3z = 0?
(A) (1, 0, 2) and (2, 1, 1)
(D) (−1, 0, 2) and (−1, 1, 1) (B) (3, 1, 2) -3- (C) (3, 0, 2) and (2, 1, 1)
(E) none of these See Next Page 4CCM113A
A 8. For which values of λ are the vectors u = 2λi − j + k, v = 4i − 3λj
− 3k and
w = −4λi + 3j − 3k a basis in R3 ?

8. (A) λ 6= 0
(B) λ 6= −1
(C) λ 6= 5
(D) never
(E) none of these A 9. Let R3 be equipped with the standard or dot
inner product. What is the
volume of the (solid) parallelepiped determined by the vectors v1 =
(1, 1, 1),
v2 = (2, −1, 2) and v3 = (1, −1, 2)?
(B) −2 (A) 3 (C) 4 (D) 7 (E) none of these A 10. What is the
(smallest) distance of the point A = (−1, 1, −2) from the line at
which the planes with coordinate equations 2x − 3y + z = 1 and x − y
+ z = 2
intersect?

9. √
√
√
√
(A) 2 3
(B) 11
(C) 3
(D) 2 5
(E) none of these A 11. For which values of λ do the vectors v1 = (1,
−1, 2), v2 = (1 − λ, −2, λ − 1) and
v3 = (−3, 1, 2) span a 2-plane in R3 ?

10. (A) λ = −3 A 12. (B) λ = 5 (C) λ = −7 (D) λ = −6 (E) none of these −3
3
3 Let A = 2 −2 −2 .
3
2
1
What are the eigenvalues of the matrix
√ A?
(A)
(B) 5, −2, −2 − 14
√ 0, 2, −2

11. (D) 12, 2, −2 -4- √
√
(C) 0, −2 + 12, −2 − 12
(E) none of these See Next Page 4CCM113A A 13. 1
2 −2
1 10
1 −2
3 . Let B = 4 −2 −3 −4
20 −1
1

12. 4
Give the determinant of the matrix B 2 .
(A) 5 A 14. (C) 16 (D) 36 (E) none of these Consider the linear map T
: R3 → R3 given by T ((x, y, z)) = (x + y, x − z, x −
2y−z). Find the matrix associated to T with respect to the basis u1 =
(1, −1, 1),
3
u2 = (1, 1,
(−2, 0, −1) in R . 1) and u3 = 1
1
−2 −3
2 −3

13. 1
−1 −3
2
2 A1 = 1 −6 1 − 21 1 − 12 , A2 = 3 9 , A3 = 5
,
2 −4
1
2 −4
1
2 −7
1 1

14. 2 −3
2 1 A4 = 2 −3 − 2 ,
2 −4
1
(A) A1 A 15. (B) 49 (B) A2 (C) A3 (D) A4 (E) none of these Give the
value y(2) of the solution to the differential equation
dy
d2 y
−
2
− 15y = 30x2 − 7x + 9 ,

15. dx2
dx
which satisfies the boundary conditions y(0) = 0 and
(A) −5e−6 + 2e10 − 7
(B) −7e−6 + e10 − 7
(D) −e8 + 2e−7 − 11
(E) none of these -5- dy
(0)
dx = 14. (C) 9e−6 + 2e10 − 7 See Next Page 4CCM113A
SECTION B
In section B you must provide FULL DERIVATIONS of your
solutions to gain full marks. Question B16 carries 25.0 marks. B 16.

16. Let V and W be vector spaces over the real numbers.
1. Give the definition of a linear map φ : V → W .
2. Consider the set L of all linear maps φ : V → W . Show that L is a
vector
space with respect to the addition
(φ1 + φ2 )(v) = φ1 (v) + φ2 (v)
and scalar multiplication
(λφ)(v) = λφ(v)
where φ1 , φ2 , φ ∈ L, v ∈ V and λ ∈ R. Verify in detail all 10
properties required
for L to be a vector space.

17. 3. If dim V = n and dim W = m, state without proof what the
dimension of
L is. -6- Final Page