# Homework of advanced Algebra

**18.02 Notes, Exercices and Solutions** are for sale at the Copy
Technology

Center in the basement of Building 11. This is where to find the exercices

labelled 1A, 1B, etc. Problem Sets have two parts , A and B.

**Part A **has problems from the text, with answers to many in the back of

the text, and problems from the Notes with solutions at the end of the Notes.

Look at the solutions if you get stuck, but try to do as much as possible
without

them. Part A will be graded quickly, checking that the problems are there and

the solutions not merely copied.

**Part B** consists of unsolved problems, is worth more points, and will
be

graded more carefully. Many of these problems are longer multi-part exercises

posed here because they do not fit conveniently into an exam or short-answer

format.

Collaboration on problem sets is encouraged, but

**1. Attempt each part of each problem yourself.** Read each portion of

the problem before asking for help. If you dont understand what is being

asked, ask for help interpreting the problem and then make an honest

attempt to solve it .

**2. Write up each problem independently.** On both Part A and B exercises

you are expected to write the answer in your own words.

**3. Write on your problem set whom you consulted and the sources
you used.** If you fail to do so, you may be charged with plagiarism and

subject to serious penalties.

**4. It is illegal to consult materials from previous semesters .**

**Reading**

The material for this problem set is covered in sections D and M of the 18.02

Notes, Exercises and Solutions, and sections 17.3, 18.1, 18.2, 18.3, 18.4 of
Simmons,

Calculus with Analytic Geometry, 2nd Edition.

**Part A **(10 points)

Exercises from the Supplementary Notes and Exercises (20 problems, 1/2 point each):

• Vectors (1A): 1A-1, 1A-2, 1A-3, 1A-5,

• Dot Product (1B): 1B-1, 1B-2, 1B-4, 1B-7,

• Determinants (1C): 1C-1, 1C-2, 1C-3,

• Cross Product (1D): 1D-1, 1D-2, 1D-3,

• Equations of Lines and Planes (1E): 1E-1,

• Matrix Algebra (1F): 1F-3, 1F-4,

• Solving Square systems ; Inverse Matrices (1G): 1G-1,

• Cramer’s Rule; Theorems about Square Systems (1H): 1H-3, 1H-5

**Part B **(25 points)

There are 5 problems worth 5 points each

1. (5 points) Label the four vertices of a parallelogram in counterclockwise

order as OPQR . Prove that the line segment from O to the midpoint of

PQ intersects the diagonal PR in a point X that is 1/3 of the way from P

to R. (Hint: Let
,
and express everything in terms

of
)

2. (5 points) The Cauchy-Schwarz inequality

(a) Prove from the geometric definition of the dot product the following

inequality for vectors in the plane or in space:

Under what circumstances does equality hold?

(b) If the vectors are plane vectors, what does this inequality say in terms

of the components of
along the vectors? If the vectors

are space vectors, what does it say in terms of the components of

and
along the vectors

(c) Give a different proof of this inequality by considering, for fixed

and
,
the function

(Hint: First, use the algebraic properties of dot product to write f

as a quadratic polynomial in t . Show that f(t) is never negative.

Thus f(t) can have at most one zero. The quadratic formula implies

that the coefficients of a quadratic polynomial with at most one zero

satisfy a certain inequality.)

3. (5 points) Using vector methods , show that the general formula for the

distance from a point (x_{0}, y_{0}, z_{0}) to the plane defined by the equation

is

4. (5 points) Consider the matrix

(a) What happens when a vector is multiplied by
that is, what is

the geometric relationship between a plane vector
Draw

a picture showing the two cases

(b) Consider , the same
matrix as (5) but with −θ in place of θ. Compute

the matrix multiplication algebraically.
Deduce immediately

(without further calculation ) a formula for the inverse matrix

. What is a geometric justification for your
conclusion?

(c) Use trigonometric identities and matrix multiplication
to show that

What is a geometric justification for this
identity?

5. (5 points) Suppose that A is a matrix with determinant
zero: |A| = 0,

and that is a solution of
Show that any other solution of this

system may be written as where
is a solution of the system

Apply this principle in the particular case of the system

to find a general formula for all solutions of this system (6).

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