Determine the general (complex) solution to the following systems: (a) 2 x+(1+i) y-2 i z=2 i

Determine the general (complex) solution to the following...

Key Concepts

Parametric (General) Solution

When a system of equations has free variables, its general solution is expressed in a parametric form. This involves writing the solution as the sum of a particular solution and a linear combination of vectors that form a basis for the null space of the system. This approach clearly encapsulates the entire solution set by incorporating arbitrary parameters for the free variables.

Gaussian Elimination

Gaussian elimination is a fundamental method for solving systems of linear equations. By systematically applying elementary row operations to an augmented matrix, one can simplify the system into an equivalent, simpler form (often row echelon or reduced row echelon form) to identify solutions or determine the presence of free variables. This step-by-step process is equally applicable when the entries of the matrix are complex.

Systems of Linear Equations with Complex Coefficients

A system of linear equations with complex coefficients involves solving for variables that may take on complex values. The techniques used are similar to those in the real case, but require careful handling of complex arithmetic. The solution process involves ensuring consistency and finding conditions under which the system is either determined, underdetermined, or inconsistent.

Complex Numbers

Complex numbers extend the real numbers by including the imaginary unit i, where i² = -1. They are essential when coefficients or solutions in equations involve numbers that cannot be represented on the real number line. Understanding complex numbers enables arithmetic operations and algebraic manipulation that are crucial in solving equations with complex coefficients.

Transcript

00:01 All right, so we're going to solve the following systems with three variables, and i am going to use the elimination method to do so.

00:09 So to do the elimination method, i need to create two new equations with only two variables, and then solve from there using the elimination method.

00:18 So i need to create an equation four and an equation five.

00:23 And i've just numbered the first three equations so you can kind of keep track of what we've done.

00:28 So looking at our variables here, i want to try to find something that's easy to cancel out.

00:34 What i think i'm going to go with, and it's not wrong to cancel out a different one, but i think i'm going to eliminate my x's because i notice one's 2x and the other one is x and a negative x.

00:48 So that should be easier to cancel out.

00:50 The ys wouldn't be as bad as well because they're 2y, 2y, and y.

00:55 So that wouldn't be as hard to cancel out.

00:56 And even the zs in this case aren't hard to cancel out.

00:59 Really how you want to start it to cancel it out.

01:02 So i'm going to create equation four by going equation two plus equation three because i notice they both have the same amount of x as one's positive, one's negative.

01:18 So negative x plus x is zero, that's gone.

01:22 Y plus 2y is 3y, 2z plus 4z is 6 z and 3 plus 0 .0 is 3.

01:31 And 3 plus 0 .000 is.

01:31 Three.

01:32 So when i go to create my next equation i have to make sure i cancel out my x's as well and i must use equation one to create it.

01:40 So i can either combine equation two or equation three.

01:45 I let's use equation two.

01:48 So i'm going to have to take equation one and i'm going to have to add two times equation two in order to cancel that out.

01:58 So i have to double my second equation and then add it.

02:01 So negative x doubled is negative 2x plus 2x is 0 y doubled is 2y plus 2y is 4y 2z doubled is 4 z plus z is 5 z and 3 doubled is 6 plus 1 is 7 so now i have created two new equations with two variables and i can use elimination again to cancel out one of those variables so i'm creating a sixth equation that is down to a single variable.

02:37 So i can either cancel out my ys or my zs.

02:40 To do this in either case, i have to manipulate both equations.

02:45 So it's not really easier to do one over the other.

02:48 So why don't we go four times my fourth equation and we will subtract three times my fifth equation just to get our ys to cancel out.

03:04 And again, it's not easier to do one over the other.

03:07 So i will write it out here just so you can kind of see what we've done.

03:10 So 3 times 4 is 12.

03:15 6 times 4 is 24.

03:18 So there's our fourth equation times by 4.

03:23 And we're subtracting 5th equation times by 3.

03:28 So 4 times 3 is 12.

03:31 5 times 3 is 15.