For Exercise 115-120, factor the expressions over the set of complex numbers. For assistance, consider these examples. · In Section R .3 we saw that some expressions factor over the set of integers . For example: x 2 − 4 = x + 2 x − 2 . · Some expressions factor over the set of irrational numbers . For example: x 2 − 5 = x + 5 x − 5 . · To factor an expression such as x 2 + 4 , we need to factor over the set of complex numbers . For example, verify that x 2 + 4 = x + 2 i x − 2 i . a . x 2 − 11 b . x 2 + 11
For Exercise 115-120, factor the expressions over the set of complex numbers. For assistance, consider these examples. · In Section R .3 we saw that some expressions factor over the set of integers . For example: x 2 − 4 = x + 2 x − 2 . · Some expressions factor over the set of irrational numbers . For example: x 2 − 5 = x + 5 x − 5 . · To factor an expression such as x 2 + 4 , we need to factor over the set of complex numbers . For example, verify that x 2 + 4 = x + 2 i x − 2 i . a . x 2 − 11 b . x 2 + 11
Solution Summary: The author explains the factors of the expressions given below over the set of complex numbers.
For Exercise 115-120, factor the expressions over the set of complex numbers. For assistance, consider these examples.
·
In Section R
.3 we saw that some expressions factor over the set of integers
. For example:
x
2
−
4
=
x
+
2
x
−
2
.
·
Some expressions factor over the set of irrational numbers
. For example:
x
2
−
5
=
x
+
5
x
−
5
.
·
To factor an expression such as
x
2
+
4
, we need to factor over the set of complex numbers
. For example,
verify that
x
2
+
4
=
x
+
2
i
x
−
2
i
.
a
.
x
2
−
11
b
.
x
2
+
11
Combination of a real number and an imaginary number. They are numbers of the form a + b , where a and b are real numbers and i is an imaginary unit. Complex numbers are an extended idea of one-dimensional number line to two-dimensional complex plane.
A factorization A = PDP 1 is not unique. For A=
7 2
-4 1
1
1
5 0
2
1
one factorization is P =
D=
and P-1
30
=
Use this information with D₁
=
to find a matrix P₁ such that
-
-1 -2
0 3
1
-
- 1
05
A-P,D,P
P1
(Type an integer or simplified fraction for each matrix element.)
Matrix A is factored in the form PDP 1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace.
30 -1
-
1 0 -1
400
0
0 1
A=
3 4 3
0 1 3
040
3 1 3
0 0
4
1
0
0
003
-1 0 -1
Select the correct choice below and fill in the answer boxes to complete your choice.
(Use a comma to separate vectors as needed.)
A basis for the corresponding eigenspace is {
A. There is one distinct eigenvalue, λ =
B. In ascending order, the two distinct eigenvalues are λ₁
...
=
and 2
=
Bases for the corresponding eigenspaces are {
and ( ), respectively.
C. In ascending order, the three distinct eigenvalues are λ₁ =
=
12/2
=
and 3 = Bases for the corresponding eigenspaces are
{}, }, and {
respectively.
Elementary Statistics: Picturing the World (7th Edition)
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