Note:

This is not yet complete; will take me a little more time to type everything up nicely. I've typed in some already, so you have an idea of how you did. Excuse mistakes in typing, will be fixed up soon.

Math 112 Final Exam

09.00 - 12.00, Thursday 19th June 1997

Instructor: Helena Verrill

NAME: Imaginary Imelda

STUDENT NUMBER: i+3

Directions:

Write your name and student number above.

Make sure your paper has 10 sheets and 6 questions.

Write all your answers to questions asked in the spaces provided. Show your working in the space below the question. Show enough working to show that you understand the methods of the course, and that you're not guessing the answers.

Question	Value	Mark
1	10	10
2	5	5
3	5	5
4	10	10
5	10	10
6	10	10
Total	50	50

1

Do the following vectors span R³?

Explain how to write this problem in the form

Explanation:

To span R3 means we can use these vectors to get to anywhere in R3, so, for any point (x,y,z) in R3, we need to be able to find a,b and c so that / 1 \ / 1 \ / 2 \ / x \ a | 3 | + b | 2 | + c | 1 | = | y | \ 1 / \ 1 / \ 1 / \ z / This is the same as solving the equation: / 1 1 2 \ / a \ / x \ | 3 2 1 | | b | = | y | \ 1 1 1 / \ c / \ z / This is Aa=x where A = a=(a,b,c) x=(x,y,z) We can always solve this only if A has an inverse, and then the solution for (a,b,c) is given by a=A-1x So These vectors span R3 if and only if the matrix A has an inverse.

Write the entries of the matrix A in the spaces provided below:

Calculate the inverse of A, and write the entries of the matrix A^-1 in the spaces provided below:

Working:

2.

In the spaces provided below, write down the matrices A, B, C, that describe

A:	B:	C:

Calculate the matrices AB, BC, and ABA

AB:	BC:	ABA:

In the following graphs, draw the images of the triangle with vertices at

Working:

3.

Which of the following set of vectors span R²? Which are linearly independent? Which form a basis for R²? Write "Y" for "yes" and "N" for "no" in the spaces provided in the table:

Working:

4.

The matrix A is given by:

Fill in the blanks:

Rank of A	2
Nullity of A	2
Basis for the column space of A
Basis for the null space of A

Working:

5.

A lecturer notices that of the people that are taking the course, not everyone attends all the lectures. Observation shows that 10% of the people at a lecture do not come to the next lecture, but of the people that do not attend, 50% do attend the next lecture.

If there are 50 people who register for the course and all of them come to the first lecture, how many people are expected to come to lecture 48, the final lecture?

Working:

Express situation as linear equations.[1 mark]

Write as an equation xn=Axn-1[1 mark]

Find eigen values of A[2 marks]

Find eigen vectors of A[2 marks]

Write x1 in terms of the eigen vectors of A[2 marks]

Find x48, and the number of people in the final lecture.[2 marks]

6.

A polynomial of degree 2 in x is an equation of the form

The set of all polynomials of degree 2 forms a vector space, called P₂.

Let B be the basis

Complete the following table:

One of the following sets is a subspace of P₂, and the other is not:

• The set of all polynomials f in P₂ with f(1)=3
• The set of all polynomials f in P₂ with f(1)=f(3)

For the set that is not a subspace, give a reason why not:

The set of polynomials with f(1)=3 is not a subspace. For example, f(x)=x2+2 satisfies f(1)=1+2=3, but 2f(x)=2x2+4 has f(1)=2+4=6, so it's not in the set of polynomials with f(1)=3

Let H be the set that is a subspace. Check that

The set of polynomials with f(1)=f(3) is a supspace. Check of (i) if f(1)=f(3), and g(1)=g(3), then for f+g, (f+g)(3)=f(3)+g(3)=f(1)+g(1)=(f+g)(1), so f+g also satisfies the property. Check of (ii) IF f(1)=f(3), then for a real number k, kf(3)=kf(1), so the property also holds for kf.

Find a basis for H. What is the dimension of H?

Basis of H
dimension of H

Working:

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