Macquarie University, Department of Mathematics and Statistics

MATH 2010/2055 — Assignment — Session 2 2021

Due 10:00pm Friday October 29

Assignment is worth 20% of you final mark. The maximum number of points is 30. This includes 15 points for calculus part and 15 points for algebra. Please make sure that your submission is readable. We are not able to mark illegible parts of your solutions.

Assignment. Algebra Part.

1. (5 marks) Consider the following matrix

(a) Use Gaussian elimination to write A in reduced row echelon form.

(b) Determine the dimension of the null space of A.

(c) Describe the space of all homogeneous solutions of the system Ax = 0 and find a basis for the null space of A.

(d) Find a base for the egigenspace corresponding to eigenvalue = 3 for the matrix

.

(e) Find all eigenvalues for the above matrix B.

2. (5 marks) Let B = {b1,b2,b3} and C = {c1,c2,c3} be two bases of R3, where

b ,

and

c .

(a) Determine [c3]B, the coordinates of c3 in the basis B.

(b) Find the transition matrix PC!B so that PC!B[x]C = [x]B. Recall that PC!B is given by

PC!B = ([c1]B [c2]B [c3]B) .

(c) Determine [b2]C, the coordinates of b2 in the basis C.

(d) Find the transition matrix PB!C so that PB!C[x]B = [x]C. Recall that PB!C is given by

PB!C = ([b1]C [b2]C [b3]C) .

1

(e) Multiply matrices PB!C and PC!B to check that

PB!CPC!B = I

3. (2 marks) Suppose that B = {b1,b2} is a basis of R2 and that the vectors b1 and b2 are eigenvectors of a linear transform T with eigenvalues a and b respectively. Calculate [T]B.

4. (3 marks)

(a) Show that the set V = {v1,v2}, where

.

forms an orthonormal basis for R2.

(b) Calculate the inner products hv1,xi andhv2,xi where .

(c) Use point (b) to calculate the coordinates [x]V .

See next page for the calculus part.

Assignment. Calculus Part.

1. [2 marks] In each of the following: if there is a limit, calculate it; if there is no limit, explain why. Use any technique of your choice.

;

.

[Hint: Revise the LíHopitalís rule]

2. [4 marks] Consider the function f : R2 !R; given by

0) ; 0) :

(a) Find the partial derivatives and

(b) Use principle deÖnition of the partial derivatives to Önd and

@x

(0;0).

(c) Calculating the appropriate limit determine whether f is di§erentiable at (0;0).

(d) Does the linear approximation of f exist in the vicinity of (0;0)? Explain.

3. [3 marks] Let the real-valued function f (x;y) of two variables be deÖned by

f(x;y) = 8y3 + 12x2 ! 24xy:

(a) Calculate the Örst partial derivatives of f.

(b) Calculate the second partial derivatives of f.

(c) Locate the stationary points of f and classify them.

4. [3 marks] Use Lagrange multipliers to Önd the point on the paraboloid z = x2 + y2 which is closest to the point (3;!6;4). Then calculate the perpendicular distance (minimum distance) from the point (3;!6;4) to the paraboloid.

1

[Hint: Find Örst the minimum of squared distance subject to the constraint]. Remark: Solving this problem you will derive the cubic polynomial equation with real coe¢cients for the unknown !. Find one root by trial and error approach.

5. [3 marks] Sketch the region of integration. Then changing the order of integration evaluate the integral: Z Z sin-y2# dy dx:

1 1

0 x

2

MATH 2010/2055 — Assignment — Session 2 2021

Due 10:00pm Friday October 29

Assignment is worth 20% of you final mark. The maximum number of points is 30. This includes 15 points for calculus part and 15 points for algebra. Please make sure that your submission is readable. We are not able to mark illegible parts of your solutions.

Assignment. Algebra Part.

1. (5 marks) Consider the following matrix

(a) Use Gaussian elimination to write A in reduced row echelon form.

(b) Determine the dimension of the null space of A.

(c) Describe the space of all homogeneous solutions of the system Ax = 0 and find a basis for the null space of A.

(d) Find a base for the egigenspace corresponding to eigenvalue = 3 for the matrix

.

(e) Find all eigenvalues for the above matrix B.

2. (5 marks) Let B = {b1,b2,b3} and C = {c1,c2,c3} be two bases of R3, where

b ,

and

c .

(a) Determine [c3]B, the coordinates of c3 in the basis B.

(b) Find the transition matrix PC!B so that PC!B[x]C = [x]B. Recall that PC!B is given by

PC!B = ([c1]B [c2]B [c3]B) .

(c) Determine [b2]C, the coordinates of b2 in the basis C.

(d) Find the transition matrix PB!C so that PB!C[x]B = [x]C. Recall that PB!C is given by

PB!C = ([b1]C [b2]C [b3]C) .

1

(e) Multiply matrices PB!C and PC!B to check that

PB!CPC!B = I

3. (2 marks) Suppose that B = {b1,b2} is a basis of R2 and that the vectors b1 and b2 are eigenvectors of a linear transform T with eigenvalues a and b respectively. Calculate [T]B.

4. (3 marks)

(a) Show that the set V = {v1,v2}, where

.

forms an orthonormal basis for R2.

(b) Calculate the inner products hv1,xi andhv2,xi where .

(c) Use point (b) to calculate the coordinates [x]V .

See next page for the calculus part.

Assignment. Calculus Part.

1. [2 marks] In each of the following: if there is a limit, calculate it; if there is no limit, explain why. Use any technique of your choice.

;

.

[Hint: Revise the LíHopitalís rule]

2. [4 marks] Consider the function f : R2 !R; given by

0) ; 0) :

(a) Find the partial derivatives and

(b) Use principle deÖnition of the partial derivatives to Önd and

@x

(0;0).

(c) Calculating the appropriate limit determine whether f is di§erentiable at (0;0).

(d) Does the linear approximation of f exist in the vicinity of (0;0)? Explain.

3. [3 marks] Let the real-valued function f (x;y) of two variables be deÖned by

f(x;y) = 8y3 + 12x2 ! 24xy:

(a) Calculate the Örst partial derivatives of f.

(b) Calculate the second partial derivatives of f.

(c) Locate the stationary points of f and classify them.

4. [3 marks] Use Lagrange multipliers to Önd the point on the paraboloid z = x2 + y2 which is closest to the point (3;!6;4). Then calculate the perpendicular distance (minimum distance) from the point (3;!6;4) to the paraboloid.

1

[Hint: Find Örst the minimum of squared distance subject to the constraint]. Remark: Solving this problem you will derive the cubic polynomial equation with real coe¢cients for the unknown !. Find one root by trial and error approach.

5. [3 marks] Sketch the region of integration. Then changing the order of integration evaluate the integral: Z Z sin-y2# dy dx:

1 1

0 x

2

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