MATH2061: Linear Mathematics and Vector Calculus
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Semester 1 2009
MATH2061: Linear Mathematics and Vector Calculus
SECTION A: Linear Mathematics
Use separate writing booklets for sections A and B.
1. (a) Complete the following definition:
The set of vectors {v1 , v2 , . . . , vn } in a vector space V is a linearly
independent set if ..................
(b) Determine whether or not W = 1 # , # , #$ is a linearly independent set in R3 .
(c) Let A = 2 1 −4 .
(i) Find the null space of A.
(ii) What is the dimension of the column space of A?
(d) Show that the set X = # ∈ R3 | x − 2y +5z = 3$ is not closed
under addition.
(e) The set Y = # ∈ R3 | x +2y − 4z = 0$ is a subspace of R3 . Find
a basis for Y .
2. (a) Let X = {(2(2) )} ∈ R2 . Give a geometric description of Span(X).
(b)
Let A = )1(2) 0(3)* .
(i) Find the eigenvalues of A.
(ii) Find a matrix P and a diagonal matrix D such that A = PDP −1 . (iii) Find A20 .
(iv) Find a formula, in terms of n, for xn if
xn+2 = 2xn+1 +3xn , with x0 = 0 and x1 = 4.
3. (a) Let M = 1 1 .
(i) Find the eigenspace of M corresponding to the eigenvalue − 1.
for all ( y(x) ) ∈ R2 .
Which straight lines through the origin in R2 are fixed by T?
(b)
Let v1 = 1 and v2 = 4(3) .
(i) Explain why {v1 , v2 } is a basis for R2 .
(ii) Write (0(1) ) as a linear combination of v1 and v2 .
(iii) Suppose that S : R2 → R2 is a linear transformation, and S(v1 ) = (5(3) ) and S(v2 ) = ( 2 ). Find S ((0(1) )).
(c) Let X = {f ∈ F | f(x) = A + Bex , A,B ∈ R}.
Determine whether or not X is a subspace of F. Justify your answer.
(d) Define a function g : R3 → R by g "" ## = x + y for all " # ∈ R3 . Is g a linear transformation? Justify your answer.
4. (a) Let L = 00(3)5 be the Leslie matrix for a particular species of insect.
Find the unique positive eigenvalue of L, and a corresponding eigen- vector.
(b) Let Y = {p1 ,p2 ,p3 } be a subset of P3 , with p1 (x) = x− 2, p2 (x) = x3 − x
(i) Show that Y is a linearly independent set. (ii) Is Y a basis for P3 ? Justify your answer.
(c) (i) Suppose λ 1 and λ2 are two different eigenvalues of a matrix A, and that v1 and v2 are corresponding eigenvectors.
Prove that there is no real number α such that v1 = αv2 .
(ii) Prove that any three eigenvectors v1 , v2 , v3 corresponding to three distinct eigenvalues λ 1 , λ2 , λ3 of a matrix A are linearly indepen- dent.
SECTION B: Vector Calculus
Use separate writing booklets for sections A and B.
1. (a) Let F = (x − 3z)i +(x + y + z)j +(x − 3y) k.
(i) Calculate ∇× F.
(ii) Is F a conservative field? Give a reason for your answer.
(iii) Evaluate +C F · d r, where C is the straight line segment from
(1, 0, 0) to (1, 2, 1).
(b) G = 3x2 i +2yz j + y2 k is a conservative vector field.
(i) Find a potential function for G.
(ii) Hence (or otherwise) find the work done by G along the curve with vector equation r(t) = ti +2 j + t2 k for t : 0 → 1.
2. (a) Evaluate
++R (4y2 − 5x)dA
where R is the triangular region bounded by the x-axis and the lines x = 1 and y = x.
(b) Hence (or otherwise) find the work done by the force field F = (x2 +5xy)i +4xy2 j
in moving a particle from the origin along the boundary of the region in part (a) taken once in an anticlockwise direction.
(c) Use polar coordinates to evaluate
+−2(2) +0 √4−y2 (x2 + y2 )3/2 dxdy .
3. (a) Let S be the section of the sphere x2 + y2 + z2 = 1 in the first octant, bounded by the planes y = 0 and y = x. Find the flux of F = z2 k across S in the outward direction.
(b) The solid half-cylinder x2 + y2 = 1, y ≥ 0, 0 ≤ z ≤ 2 has density (1 + x2 + y2 )−1 at any point (x, y, z). Calculate its mass.
4. (a) Let F = x3 i + y3j + zk.
Use the Divergence Theorem to calculate the flux of F outwards across the closed surface bounded by x2 + y2 = 1, z = 0 and z = x +2.
(b) Evaluate 0 1 x 1 xey3 dy dx.
(c) Given
F = (2x + sin yz)i +(2y + xz cosyz)j +(y2 +2z + xy cosyz)k,
evaluate C F . dr where C is the curve C consisting of the straight
line segment from (0, 0, 0) to (0, 1, 0), followed by the quarter circle y2 +z2 = 1 from (0, 1, 0) to (0, 0, 1), followed by the straight line segment from (0, 0, 1) to (0, 0, 0).
2022-06-07