闪电代写 -代写CS作业_CS代写_Finance代写_Economic代写_Statistics代写_代码代做_IT代写_加急帮助

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Summer Assessment, 2020

School of Mathematics and Statistics

MAST10007 Linear Algebra

Question 1 (6 marks)

(a) Let the ages (in years) of three people Frank, Dharma and Phil be denoted F , D and P respectively. You are told that the sum of Dharma’s and Phil’s ages is 13 more than Frank’s age. You are further told that Frank’s age plus Phil’s age is 19 more than Dharma’s age, and that the sum of all three ages is 71.

(i) Specify three linear equations implied by the above information.

(ii) Write the equations in part (i) as an augmented matrix and solve to determine the three ages.

(b) Consider the linear system in the real variables

x1 , x2 , x3c

x2 + x3 = 2

2x2 + kx3 = k,

where k e R. Determine the values of k such that the system has

(i) no solutions

(ii) a unique solution.

Question 2 (6 marks)

(a) If A = ┌ 1_1┐

(i) AT A

(ii) AAT .

(b) Let X be an invertible matrix. Show that

det(X← 1 ) = 1

D1 = det ┌c(a) d(b)┐ ,

where a, b, c, d e R.

(i) Draw a ﬁgure involving the vectors (a, b) and (c, d) that has area |D1|.

(ii) Express

D2 = det ┐

in terms of D1 . Justify your answer.

Question 3 (6 marks)

(a) Let

A = ┌1(0) 0(0) !1 1

2(1)┐

2! .

(i) Using elementary row operations, compute A ← 1 .

(ii) If

AB = ┌ ┐

! !

calculate B .

(iii) Is

Explain

A = ┌ 5(2) !5

your answer.

2(1)┐ B ← 1 ?

(b) Given that

┐ R2 - R2 + 2R1 ~ ┌ 0(1) 1(1) ┐ R1 - R1 _ R2 ~ ┌0(1) 1(0)┐

write

┐ ← 1

as a product of elementary matrices.

Question 4 (6 marks)

Consider the line L given by

L : r = (1, 1, _1) + t(2, 3, 1), t e R.

(a) Determine the Cartesian form of L.

(b) Find the Cartesian form of the plane containing L and the line

r = (1, 1, _1) + s(1, _1, 1), s e R.

R = {t(1, 1, _2) + s(1, _1, 1) : 0 < t < 1 and 0 < s < 1}.

Question 5 (6 marks)

(a) Consider the subset of M2|2 deﬁned by

H = ┐ : b e R、.

(i) Use the subspace theorem to show that H is a subspace of M2|2 .

(ii) What is the dimension of H? Explain your answer.

(b) Consider the subset of R3 deﬁned by

S = {(x, y, z) : xy + xz + yz = 0}.

Show that S is not a subspace of R3 .

Question 6 (8 marks)

(a) Let x1 , x2 , x3 e R4 . Suppose that x2 0, x1 cx2 for all c e R, and

2x1 _ x2 + x3 = 0.

What is the dimension of span {x1 , x2 , x3 }? Justify your answer.

(b) Let A be a 4 × 5 matrix and let v1 , v2 , v3 be column vectors with 4 entries. Suppose that

┌ ┐ ┌ A v1 v2 v3 ┐ ~ ))! .

(i) Denote the columns of A by a1 , . . . , a5 in order. Write v3 as a linear combination of the columns of A.

(ii) Which of the vectors v1 , v2 , v3 belong to the column space of A? Give a reason. (iii) Determine a basis for the solution space of A.

(i) Give a geometrical description of the column space of B . (ii) Give a geometrical description of the solution space of B .

Question 7 (6 marks)

(a) Let T : R2 - R2 be a linear transformation. Suppose that T (1, 0) = (1, 1) and T (0, 1) = (_1, 1).

(i) Illustrate on a diagram how T maps the unit square with corners (0, 0), (1, 0), (1, 1) and (0, 1), and describe T geometrically.

(ii) Specify the standard matrix of T ← 1 .

(b) Deﬁne the linear transformation S : M2|2 - M2|1 by

S ╱ ┌c(a) d(b)┐ 、= ┐ .

(i) What is the size of the standard matrix [S]?

(ii) Determine [S].

Question 8 (6 marks)

(a) Let b = (1, 1, 1, 1) and deﬁne T : R4 - R4 by

T (x) = (x . b)b,

where . is the dot product. Show that T is a linear transformation by verifying the two deﬁning properties.

(b) Let S : R3 - R3 be the linear transformation mapping each vector x e R3 to its orthogonal projection onto the plane x + y + z = 0.

(i) Specify a basis for the image of S .

(ii) Specify a basis for the kernel of S .

(iii) Is S invertible? Give a reason.

Question 9 (7 marks)

Let B = {b1 , b2 , b3 } where

b1 = (1, 0, 1), b2 = (1, 1, 0), b3 = (0, 0, 1).

Denote by s the standard basis in R3 .

(a) Verify that B is a basis for R3 .

(b) Given [x]毖 = !(┌)1_11!(┐), specify x.

(d) Let c = {c1 , c2 , c3 } be a basis for R3 such that

b1 = c1 + c2 , b2 = c2 + c3 , b3 = c1 + c2 + c3 .

(i) Specify the change of basis matrix PC |毖 and calculate P毖 |C .

(ii) Use parts (c) and (d)(i) to calculate P扌 |C . Hence determine the explicit form of c1 , c2 and c3 .

Question 10 (6 marks)

Let

A = ┌3(1) 1(3)┐ .

(a) Find the eigenvalues of A.

(b) Relate the eigenvalues of A to det(A).

(d) (i) Write A in the form A = QDQT for some diagonal matrix D and orthogonal matrix Q.

(ii) Specify the corresponding decomposition for A ← 1 .

Question 11 (6 marks)

A person beginning a once a week exercise program has kept the following record of the number of laps y of a 50 metre pool they have swum during the session for week x:

x y