Complex Numbers in Linear Algebra

In summary, complex numbers are defined as ordered pairs of real numbers and are denoted by C. They can be written as (a,b) where a is identified with the real number a and b is identified with i. The complex number (0,1) is denoted as i and has the property that i2 = -1. Addition and multiplication of complex numbers follow the operations of addition and multiplication of real numbers, except for the product which is defined as (a,b)*(c,d) = (ac-bd, ad+bc). This definition of multiplication leads to (0,1)*(0,1) = (-1,0) which shows that (0,1) is indeed i. Defining i as (
  • #1
cowmoo32
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I'm working my way through Shaum's Outline on linear algebra and in it they define a complex number as an ordered pair of real numbers (a,b). So given a real number a, its complex counterpart would be (a,0). Operations of addition and multiplication of real numbers work under the correspondence:

(a,0) + (b,0) = (a + b,0)
(a,0)*(b,0) = (ab,0)

I can follow that, but I'm confused how they define i. I know i=(-1)1/2. They define it as:

i2 = ii = (0,1)(0,1) = (-1,0) = -1

So am I to assume that any complex number written as (0,b) = -b?
 
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  • #2
The complex number (a,b) would be written a+bi. I think you might be getting confused because they defined i as i2 = -1, so, as you wrote, i = -11/2
 
  • #3
The next step after defining i is writing a complex number z = a + bi. I'm just a little confused as to their definition of i.

Here is the first definition of complex numbers, verbatim:
The set of complex numbers is denoted by C. Formally, a complex number is an ordered pair (a,b) of real numbers.

We identify the real number a with the complex number (a,0); that is,

a<-->(a,0)

Thus, we view R as a subset of C, and replace (a,0) by a whenever convenient and possible

The complex number (0,1) is denoted by i. It has the important property that...
and then they go on to define i in the manner I posted above. After that,

Accordingly, any complex number z = (a,b) can be written in the form

z = (a,b) = (a,0) + (0,b) = (a,0) + (b,0)*(0,1) = a + bi

Ok, so as I typed that I see that (0,1) is simply i.
 
  • #4
The one thing you did NOT say was how to multiply two such pairs- you only say "Operations of addition and multiplication of real numbers work under the correspondence:

(a,0) + (b,0) = (a + b,0)
(a,0)*(b,0) = (ab,0) "

The definition of the sum and product of two general such pairs is
(a, b)+ (c, d)= (a+ c, b+ d) ("component wise" addition)
(a, b)* (c, d)= (ac- bd, ad+ bd) (which is definitely not "component wise")

Your text should also show that all the usual "rules of arithmetic" ( addition and multiplication are associative and commutative and multiplication distributes over addition. There are additive and multiplicative identies, every pair has an additive inverse, and every pair except (0, 0) has a multplicative inverse).

From that definition of multplication (0, 1)*(0, 1)= (0*0- 1*1, 0*1+ 1*0)= (-1, 0). Since we are interpreting the pair (-1,0) to be the number "-1", that says that (0,1)*(0,1)= (0, 1)2= -1 and so (0, 1) is i. Given that we can then say that (a, b)= (a, 0)+ (0, b)= a(1, 0)+ b(0, 1)= a+ bi.

By the way, here we are defining i to be (0, 1), and then showing that i2= -1, not showing that (0, 1)= (-1)1/2. There are technical problems with "defining" i by "i=(-1)1/2". Every complex number, like every real number, has two square roots. Which of the two roots of -1 is "i"? The real numbers form an "ordered field" so we can distinguish between positive and negative square roots of numbers but the field of complex numbers is NOT an ordered field so it is impossible, a priori, to distinguish between the two roots of -1. Defining i to be the pair (0, 1) avoids that problem. (The other root of -1 is, of course, (0, -1)= -(0, 1)= -i. But to be able to call it "-i" we must be able to first distinguish that root from "i" itself.)
 
  • #5


I would like to clarify and expand upon the concept of complex numbers in linear algebra. Complex numbers are a fundamental part of linear algebra and are represented by the set of numbers of the form (a, b), where a and b are real numbers and i is the imaginary unit defined as i = √(-1). Complex numbers are often used to represent quantities that cannot be expressed using only real numbers, such as solutions to quadratic equations.

In linear algebra, complex numbers are used to represent vectors and matrices. The real part of a complex number (a, b) is denoted as Re(a, b) and the imaginary part is denoted as Im(a, b). For example, the complex number (3, 4) has a real part of 3 and an imaginary part of 4.

The operations of addition and multiplication for complex numbers follow the same rules as for real numbers, with the added consideration of the imaginary unit i. For example, (a, b) + (c, d) = (a + c, b + d) and (a, b)(c, d) = (ac - bd, ad + bc). It is important to note that complex numbers do not follow the commutative property, meaning that the order of operations matters. For example, (a, b)(c, d) ≠ (c, d)(a, b).

Now, let's address the confusion about the definition of i. The symbol i is used to represent the imaginary unit, which is defined as i = √(-1). This means that i2 = -1. However, in linear algebra, complex numbers are represented as (a, b) and the imaginary unit is represented as (0, 1). Therefore, (0, 1)(0, 1) = (0*0 - 1*1, 0*1 + 1*0) = (-1, 0) = -1. This is why i is defined as (0, 1) in linear algebra.

In summary, complex numbers are a fundamental part of linear algebra and are represented by the set of numbers of the form (a, b). The imaginary unit i is defined as i = √(-1) and is represented as (0, 1) in linear algebra. The operations of addition and multiplication for complex numbers follow the same rules as for real numbers, with the added consideration
 

Related to Complex Numbers in Linear Algebra

What are complex numbers?

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (defined as the square root of -1). The real part of the complex number is a, and the imaginary part is bi.

How are complex numbers used in linear algebra?

In linear algebra, complex numbers are used to represent vectors and matrices with complex elements. They are also used in solving systems of linear equations and in representing transformations in complex vector spaces.

What is the difference between real and complex matrices?

Real matrices have only real numbers as elements, while complex matrices can have complex numbers as elements. In addition, the operations of addition, subtraction, and multiplication for complex matrices are defined differently than for real matrices.

Can complex numbers be visualized geometrically?

Yes, complex numbers can be visualized geometrically using the complex plane, where the real axis represents the real part of the complex number and the imaginary axis represents the imaginary part. The magnitude of a complex number can also be represented as the distance from the origin to the point on the complex plane.

What is the importance of complex numbers in quantum mechanics?

Complex numbers play a crucial role in quantum mechanics as they are used to represent physical quantities such as wave functions and probability amplitudes. They also play a role in understanding the behavior of particles at the quantum level and in making predictions about their behavior.

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