Download MATH 201 Linear Algebra Homework 4 Answers

MATH 201 Linear Algebra
Homework 4 Answers
Due on or before class time on Tuesday, February 10, 2004. Be sure to check the homework format
rules on the Web site where you found this homework.
1. Read about complex numbers in the appendix to your text. Then go to Drexel’s main Web page and
select the Math Form. Type in the search phrase “complex number” and you will see lots of
possibilities. Look around and find a resource that suits you. Then answer the following questions.
(a) Write the complex number 2 + 3i in the standard form a + bi.
1–i
Use the rules of complex arithmetic:
2 + 3i = 2 + 3i 1 + i = 1 (–1 + 5i) = – 1 + 5 i <-- standard form
2 2
2
2
1–i
(b) Find the real and imaginary parts of 2 + 3i .
1–i
The results of part (a) lead to a real part of -1/2 and an imaginary part of 5/2. No i is attached to the
imaginary part, ever.
(c) Find the real and imaginary parts of (2 + i)e (–1 + 2i)t. Read about Euler’s formula in the Math Forum
database.
The strategy is to expand all parts, write in standard form, and collect the real and imaginary parts.
Using Euler’s formula,
(2 + i)e (–1 + 2i)t = (2 + i) e –t e i2t = = (2 + i) e –t (cos 2t + isin 2t)
= e –t (2cos 2t – sin 2t) + ie –t (cos 2t + 2sin 2t)
The real part is e –t (2cos 2t – sin 2t) and the imaginary part is e –t (cos 2t + 2sin 2t) . Notice that no i is
attached to the imaginary part.
(d) Find z when 2 + 3i . Note: z is sometimes called the modulus of z.
1–i
Use the simplified result in part (a) : 2 + 3i = – 1 + 5 i = 26
2 2
2
1 i
2. True or false ( give a good reason):
Short demonstrative examples are great when a statement is false.
(a) If the columns of a matrix are dependent, so are the rows.
This is false in general. The matrix 1 0 1 has linearly dependent columns and linearly independent
011
rows.
1
(b) The column space of a 2 by 2 matrix is the same as its row space.
The statement is false in general. When talking about the row space and the column space in the same
breath, it is useful to recall that the row space of A is essentially the column space of AT. This ensures
that you are comparing apple with apples and not oranges. Look at the matrix
12
12
Its column space is the subspace spanned by 1 and its row space is spanned by 1 . The two
1
2
subspaces are not the same. Make sure you are talking about subspaces and not individual vectors.
(c) The column space of a 2 by 2 matrix has the same dimension as its row space.
It is a true statement because the row space and column space have the same dimension. This common
number is called the rank of the matrix.
(d) The columns of a matrix are a basis for the column space.
This statement is not always true and so the statement is false. A familiar matrix 1 0 1 shows an
011
example. A basis consists of linearly independent vectors and the columns are linearly dependent in
this example..
3. page 269, #14.
If A is a 4x3 matrix, what is the largest possible dimension of the row space of A?
According to the work developed in class, the row rank and the column rank are identical. With A
having but 3 columns, it is apparent that the largest dimension of the row space of A is 3.
If A is a 3x4 matrix, what is the largest possible dimension of the row space of A?
This is even easier because the dimension of the row space can never exceed the number of rows,
namely 3. The number of columns does not enter into the discussion.
4. page 270, #22. Is it possible that all solutions of a homogeneous system of ten linear equations in
twelve variables are multiples of one fixed nonzero solution? Discuss.
The key word in the problem is all. Think of a linear system Ax = 0. The row reduced echelon form of
A has at most ten pivots because there are ten rows. This means there are at least two free variables
because A has twelve columns. However, the number of free variables is also dim(Nul A) and so
dim(Nul A) ≥ 2. This means that every solution of Ax = 0 is a linear combination of two linearly
independent vectors in the null space of A and one solution of Ax = 0 won’t suffice for all solutions.
A spell checker was used on this document.
2