Mathematical Background for Biostatistics Assignment 3 : Total Credit 40%

Total Marks: 80

Due: Monday, May 31, 2021

——————————————————————————–

Instructions

• Total will be scaled to contribute 40% of overall assessment.

• Show all steps.

• No half marks will be allocated to 1 mark questions.

• No mark for just writing down the formula.

• All questions and sub questions (if any) must be answered for full mark.

• Please submit your assessment item on or before the due date. If you need an extension of time to submit your assessment item, or you cannot submit electronically, please contact Murthy Mittinty (murthy.mittinty@adelaide.edu.au)

——————————————————————————–

Q1

Solve the following linear system of equations. Write the solution set in the vector form. Check your solution.

Write one particular solution and one homogeneous solution, if they exist. [3 marks]

? x + 3y = 4

??

x - 2y + z = 1 ??2x + y + z = 5

Q2

Consider the system of equations.

?x - z + 2w = -1

????x + y + z - w = 2

????5-xy+ 2- 2yz-+ 3z w+ 4=w-= 13

1

Q3

(a) Write augmented matrix for this system. [1 mark]

(b) Use elementary row operations to find its row reduced echelon form.[2 marks]

Q3

Let

2 1 M =

3 -1

(a) Calculate MTM-1. Is M symmetric? [2 marks]

(b) What is the trace of the transpose of f(M), where f(M) = M2- 1. [2 marks]

Q4

Let M be any 2 × 2 matrix. Prove that

det(M) = - tr(M2) + [tr(M)]2

[2 marks]

Q5

Let X be any n × 1 matrix subject to

XTX = (1)

. Where (1) is 1 × 1 identity matrix (i.e. a single element matrix).

Define

H = I - 2XXT

where I is an n × n identity matrix.

Show H = HT = H-1

. [4 marks]

Q6

(a) Consider the following set of vectors.

?1? ?3? ?1? ?0? ?0?

{?2?,?2?,?0?,?1?,?0?}

3 1 0 0 1

Does this set form a basis for IR3? Justify your answer. [1 mark] (b) Find a basis for IR4 that includes the vectors

?1? ?4?

???23???and??32???

?

4 1

[3 marks]

2

Q7

Q7

Use Gram - Schmidt process to find the orthonormal basis for

?1? ?1? ?0?

span{??11???,???01???,???01???}

?

1 1 2

[2 marks]

Q8

John collects the following dataset which he believes to be well approximated by a parabola y = a + bx + cx2

Table 1: John’s data

y x

5 -2

2 -1

0 1

3 2

(a) Write down the system of four linear equations for computing the unknown coefficients a,b and c.[2 marks]

(b) Write down the augmented matrix for this system of linear equations. [1 mark]

(c) Consider the left hand side of the augmented matrix to be X, a 4 × 3 matrix and y be 4 × 1 matrix of right side of the augmented matrix.

(i) Calculate XTX and XTy. [2 marks]

(ii) Calculate (XTX)-1. [2 marks]

(iii) Calculate (XTX)-1XTy. [2 marks]

You should get a 3 × 1 This is the solution of the least squares estimates for the coefficients (a,b,c). (d) Using the estimates from (c), what values does John predict for y when x = 2? [1 mark]

Q9

Using Gaussian elimination method, check if the following systems have Unique solution, infinitely many solutions or no solution.

(a)

?

2x + 4y - 3z = -1

??

5x + 10y - 7z = -2

??3x + 6y + 5z = 9

[2 marks]

(b)

? 3x - y - 5z = 9

??

y - 10z = 0

??-2x + y = -6

[2 marks]

Q10

Q10

(a) Find the rank of the matrix

?1

B = ?1

2 2

5

4 3?

9?

6

[1 mark]

(b) Check if the following matrices are symmetric or skew symmetric.

(i)

?2 1 3 ?

A = ?1 5 -3?

3 -3 7

[1 mark]

(ii)

? 0 1 3?

B = ?-1 0 2?

-3 -2 0

[1 mark]

Q11

Prove that, for any m × n matrix A, any n × p matrix B and scalars r,s, the following holds:

(rA + sB)T = rAT + sBT [3 marks]

Q12

Using Cramer’s rule, find the values of x,y,z such that

(a)

?1 2 1??x? ?1?

?3 2 1??y? = ?2?

2 -3 2 z 3

[2 marks]

(b)

( x + 2y = 1 2x - y = 2

[2 marks]

(c)

? x + 2y + z = 1

??

2x - y - z = 2

??x + z = 1

[2 marks]

Q13

Q13

(a) Decide whether

? 4 ?

v = ? 4 ?

-3

is a linear combination of the vectors

? 3 ? ? 2 ? u1 = ? 1 ?and u2 = ?-2?

-1 1

[2 marks]

(b) Decide whether

?4? v = ?4?

4

is a linear combination of the vectors

? 3 ? ? 2 ? u1 = ? 1 ?and u2 = ?-2?

-1 1

[2 marks]

Q14

(a) Determine if the following vectors are orthogonal to each other.

? 2 ? ?1? u1 = ? 1 ?and u2 = ?3?

-1 5

[2 marks]

Q15

Verify Cauchy Schwartz inequality for the following vectors

(a)

? 2 ? ?1? u1 = ? 1 ?and u2 = ?3?

-1 5

[2 marks]

(b)

? 2 ? ? 1 ? u = ? 3 ?and v = ?-2?

-4 1

[2 marks]

Q16

Q16

Let

?1? ?3?

u = ?1?and v = ?2?? IR3

0 0

Show that

?4? w = ?5?

0

is in span{u,v} [2 marks]

Q17

Check if the following vectors are linearly independent of each other.

?1? ?2? ?0? ?3?

{??23??,??10???,???11???,???22???}

? ? ?

0 1 2 0

[3 marks]

Q18

(a) Show that the set of vectors 1 -1

{u,v} = { , }

1 1

is orthogonal set of vectors. Find the corresponding orthonormal set. [2 marks]

(b) Consider the set

?1? ?3?

{u,v} = {?1?,?2?}? IR3

0 0

.

Use Gram Schmidt algorithm to find orthonormal set of vectors {w1,w2} having same span. [2 marks]

Q19

(a) Find eigenvalues and eigenvectors for

-5 2 A =

-7 4

[3 marks]

(b) Find the eigenvalues for

? 5 -10 -5?

A = ? 2 14 2 ?

-4 -8 6

[2 marks]

(c) Without writing the determinant form, identify the eigenvalues for

Q20

?1

A = ?0

0 2

4

0 4?

7?

6

. Which result did you use to arrive at a conclusion? [2 marks]

Q20

Suppose

? 2

A = ? 1

-2 0

4

-4 0 ?

-1?

4

Find an invertible matrix P and a diagonal matrix D such that P-1AP = D [3 marks]

Q21

Prove that- If S = {u1,u2,...,un} is an orthogonal set of non-zero vector space V, then S is linearly independent.[3 marks]

Total Marks: 80

Due: Monday, May 31, 2021

——————————————————————————–

Instructions

• Total will be scaled to contribute 40% of overall assessment.

• Show all steps.

• No half marks will be allocated to 1 mark questions.

• No mark for just writing down the formula.

• All questions and sub questions (if any) must be answered for full mark.

• Please submit your assessment item on or before the due date. If you need an extension of time to submit your assessment item, or you cannot submit electronically, please contact Murthy Mittinty (murthy.mittinty@adelaide.edu.au)

——————————————————————————–

Q1

Solve the following linear system of equations. Write the solution set in the vector form. Check your solution.

Write one particular solution and one homogeneous solution, if they exist. [3 marks]

? x + 3y = 4

??

x - 2y + z = 1 ??2x + y + z = 5

Q2

Consider the system of equations.

?x - z + 2w = -1

????x + y + z - w = 2

????5-xy+ 2- 2yz-+ 3z w+ 4=w-= 13

1

Q3

(a) Write augmented matrix for this system. [1 mark]

(b) Use elementary row operations to find its row reduced echelon form.[2 marks]

Q3

Let

2 1 M =

3 -1

(a) Calculate MTM-1. Is M symmetric? [2 marks]

(b) What is the trace of the transpose of f(M), where f(M) = M2- 1. [2 marks]

Q4

Let M be any 2 × 2 matrix. Prove that

det(M) = - tr(M2) + [tr(M)]2

[2 marks]

Q5

Let X be any n × 1 matrix subject to

XTX = (1)

. Where (1) is 1 × 1 identity matrix (i.e. a single element matrix).

Define

H = I - 2XXT

where I is an n × n identity matrix.

Show H = HT = H-1

. [4 marks]

Q6

(a) Consider the following set of vectors.

?1? ?3? ?1? ?0? ?0?

{?2?,?2?,?0?,?1?,?0?}

3 1 0 0 1

Does this set form a basis for IR3? Justify your answer. [1 mark] (b) Find a basis for IR4 that includes the vectors

?1? ?4?

???23???and??32???

?

4 1

[3 marks]

2

Q7

Q7

Use Gram - Schmidt process to find the orthonormal basis for

?1? ?1? ?0?

span{??11???,???01???,???01???}

?

1 1 2

[2 marks]

Q8

John collects the following dataset which he believes to be well approximated by a parabola y = a + bx + cx2

Table 1: John’s data

y x

5 -2

2 -1

0 1

3 2

(a) Write down the system of four linear equations for computing the unknown coefficients a,b and c.[2 marks]

(b) Write down the augmented matrix for this system of linear equations. [1 mark]

(c) Consider the left hand side of the augmented matrix to be X, a 4 × 3 matrix and y be 4 × 1 matrix of right side of the augmented matrix.

(i) Calculate XTX and XTy. [2 marks]

(ii) Calculate (XTX)-1. [2 marks]

(iii) Calculate (XTX)-1XTy. [2 marks]

You should get a 3 × 1 This is the solution of the least squares estimates for the coefficients (a,b,c). (d) Using the estimates from (c), what values does John predict for y when x = 2? [1 mark]

Q9

Using Gaussian elimination method, check if the following systems have Unique solution, infinitely many solutions or no solution.

(a)

?

2x + 4y - 3z = -1

??

5x + 10y - 7z = -2

??3x + 6y + 5z = 9

[2 marks]

(b)

? 3x - y - 5z = 9

??

y - 10z = 0

??-2x + y = -6

[2 marks]

Q10

Q10

(a) Find the rank of the matrix

?1

B = ?1

2 2

5

4 3?

9?

6

[1 mark]

(b) Check if the following matrices are symmetric or skew symmetric.

(i)

?2 1 3 ?

A = ?1 5 -3?

3 -3 7

[1 mark]

(ii)

? 0 1 3?

B = ?-1 0 2?

-3 -2 0

[1 mark]

Q11

Prove that, for any m × n matrix A, any n × p matrix B and scalars r,s, the following holds:

(rA + sB)T = rAT + sBT [3 marks]

Q12

Using Cramer’s rule, find the values of x,y,z such that

(a)

?1 2 1??x? ?1?

?3 2 1??y? = ?2?

2 -3 2 z 3

[2 marks]

(b)

( x + 2y = 1 2x - y = 2

[2 marks]

(c)

? x + 2y + z = 1

??

2x - y - z = 2

??x + z = 1

[2 marks]

Q13

Q13

(a) Decide whether

? 4 ?

v = ? 4 ?

-3

is a linear combination of the vectors

? 3 ? ? 2 ? u1 = ? 1 ?and u2 = ?-2?

-1 1

[2 marks]

(b) Decide whether

?4? v = ?4?

4

is a linear combination of the vectors

? 3 ? ? 2 ? u1 = ? 1 ?and u2 = ?-2?

-1 1

[2 marks]

Q14

(a) Determine if the following vectors are orthogonal to each other.

? 2 ? ?1? u1 = ? 1 ?and u2 = ?3?

-1 5

[2 marks]

Q15

Verify Cauchy Schwartz inequality for the following vectors

(a)

? 2 ? ?1? u1 = ? 1 ?and u2 = ?3?

-1 5

[2 marks]

(b)

? 2 ? ? 1 ? u = ? 3 ?and v = ?-2?

-4 1

[2 marks]

Q16

Q16

Let

?1? ?3?

u = ?1?and v = ?2?? IR3

0 0

Show that

?4? w = ?5?

0

is in span{u,v} [2 marks]

Q17

Check if the following vectors are linearly independent of each other.

?1? ?2? ?0? ?3?

{??23??,??10???,???11???,???22???}

? ? ?

0 1 2 0

[3 marks]

Q18

(a) Show that the set of vectors 1 -1

{u,v} = { , }

1 1

is orthogonal set of vectors. Find the corresponding orthonormal set. [2 marks]

(b) Consider the set

?1? ?3?

{u,v} = {?1?,?2?}? IR3

0 0

.

Use Gram Schmidt algorithm to find orthonormal set of vectors {w1,w2} having same span. [2 marks]

Q19

(a) Find eigenvalues and eigenvectors for

-5 2 A =

-7 4

[3 marks]

(b) Find the eigenvalues for

? 5 -10 -5?

A = ? 2 14 2 ?

-4 -8 6

[2 marks]

(c) Without writing the determinant form, identify the eigenvalues for

Q20

?1

A = ?0

0 2

4

0 4?

7?

6

. Which result did you use to arrive at a conclusion? [2 marks]

Q20

Suppose

? 2

A = ? 1

-2 0

4

-4 0 ?

-1?

4

Find an invertible matrix P and a diagonal matrix D such that P-1AP = D [3 marks]

Q21

Prove that- If S = {u1,u2,...,un} is an orthogonal set of non-zero vector space V, then S is linearly independent.[3 marks]

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