Linear algebra - solve linear system with complex constants

In summary, the linear system is solved by setting z = 1-y, simplifying the powers of i to -1, and solving for x and y using the equations ix + (1+i)y = i and x + y = 1.
  • #1
braindead101
162
0
Solve the following linear system:
ix + (1+i)y = i
(1-i)x + y - iz = 1
iy + z = 1


I am getting nowhere with this.
is there a trick to do these? I keep getting more and more variations of i. like i^2-1, and (1-i^2)-1

ix + (1+i)y=i
(1-i)x + y-iz=1
y + z = 1

ix + (1+i)y = i
i(1-i)x - (i^2-1)z = i-1 [iR2-r3]
[(1-i)(1+i)-i]x - (1+i)iz = (1+i)-i [(1+i)R2-R1]

okay.. and i simplified this, and got stuck.

ix + (1+i) = i
i(1-i)x + (1-i^2)z = i-1
[(1-i^2)-i]x - (1+i)iz = 1any help would be great, thanks.
 
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  • #2
Since i is the sqrt(-1) then i^2 = -1. That will simplify all of your powers of i down to simple complex numbers. Other than that there is nothing different or unusual in solving this system.
 
  • #3
Looks to me like it's straight forward. If y+ z= 1 then clearly z= 1- y. Put that into the second equation and you have (1-i)x+ y- i- iy= (1-i)x+ (1-i)y= 1- i so x+ y= 1. From that, y= 1-x so the first equation becomes ix+ (1+i)(1-x)= ix+ 1-x+i- ix= x+ 1-i= i. That should be easy to solve.
 

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their properties. It involves the use of matrices, vectors, and other mathematical concepts to solve problems related to linear systems.

2. What are complex constants in linear algebra?

Complex constants in linear algebra refer to numbers that contain both real and imaginary components. They are often represented using the symbol "i" to represent the imaginary unit, where i² = -1. In linear systems, complex constants are used to represent quantities that cannot be expressed using real numbers.

3. How do you solve a linear system with complex constants?

To solve a linear system with complex constants, we use the same methods as we would for a system with real constants. This involves setting up a system of equations, representing them using matrices, and then using techniques such as row reduction, substitution, or Cramer's rule to find the solutions.

4. Can a linear system with complex constants have multiple solutions?

Yes, a linear system with complex constants can have multiple solutions. This can occur when the system has more variables than equations, resulting in an infinite number of solutions. It can also happen when the system has inconsistent equations that lead to no solutions.

5. How is linear algebra used in real life?

Linear algebra has many applications in real life, such as in engineering, physics, computer graphics, and data analysis. It is used to model and solve problems related to linear systems, such as electrical circuits, motion of objects, and optimization of resources. It is also used in machine learning and artificial intelligence to process and analyze large amounts of data.

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