Math 112 Final Exam

09.00 - 12.00, Thursday 19th June 1997

Instructor: Helena Verrill

NAME:

STUDENT NUMBER:

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
2	5
3	5
4	10
5	10
6	10
Total	50

1

Do the following vectors span R³?

Explain how to write this problem in the form

Explanation:

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
Nullity of A
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:

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:

vector in terms of B	Polynomial in x
(19,3,0)B	19x2+3x
(1,2,3)B
	x2-7

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:

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

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

Basis of H
dimension of H

Working:

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