Use Gaussian elimination to find the complete solution to...

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists.
$$\left\{\begin{array}{ll}
2 w-x+3 y+z=0 \\
3 w+2 x+4 y-z=0 \\
5 w-2 x-2 y-z=0 \\
2 w+3 x-7 y-5 z=0
\end{array}\right.$$

Key Concepts

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations. It involves using elementary row operations to transform the augmented matrix of the system into a simpler form, typically row echelon form, so that the solution can be found through back substitution or by recognizing free variables.

Augmented Matrix Representation

An augmented matrix is a compact way to write a system of linear equations. It consists of the coefficients of the variables along with an additional column for the constants from the right-hand side of the equations. This matrix form is essential for applying systematic row operations during Gaussian elimination.

Row Echelon Form and Pivot Positions

Row echelon form is a form of the matrix where each nonzero row begins with a leading entry (pivot) and the pivot in any row is to the right of the pivot in the row above it. Identifying these pivot positions is key to understanding the structure of the solution space, such as determining which variables are leading and which are free.

Free Variables and Parametric Solutions

When the number of pivot positions is less than the number of variables, some variables do not have a leading coefficient and are referred to as free variables. These free variables can take on arbitrary values, and the complete solution to the system is typically expressed in a parametric form that describes the infinite set of solutions.

Homogeneous Systems

A homogeneous system of linear equations is one in which all the constant terms are zero. Such systems always have at least the trivial solution (where all variables are zero). Additionally, if there are free variables due to the matrix not having full rank, the system will have infinitely many solutions, which can be described in terms of the free parameters.

Transcript

00:01 In this video, i'm going to solve the system of equations in four variables.

00:06 First of all, let us rewrite it as a matrix.

00:12 So just write down all the coefficients and the right -hand side.

00:18 So that gives me 2 -1 -3 -10, 3, 2, 4 minus 1, 0, 5, minus 2, minus 2 minus 1 0, and 2, 3, minus 7, minus 5, 0.

00:40 So first of all, let us divide the first row by 2 to get a pivot.

00:49 So that will be 1 minus 1 half, 3 halves, 1 half, and 0, and keep the other parts.

01:12 Just copy the other 2 rows, the other 3 rows.

01:24 And then we want to eliminate everything below the leading one here.

01:30 I want to keep the other entries.

01:38 So in order to make 0, in the first entry of the second row, i have to subtract 3 times r1.

01:57 So that gives me 0, 7 -5s, minus 5 halves.

02:08 Following the same strategy, i can get the third row as 1⁄2 minus 19 halves and minus 7 halves.

02:25 Again, the same strategy.

02:30 I get 0, 4, minus 10, minus 6.

02:35 And then i want to divide the second row by seven halves.

02:54 So row 2 over seven halves.

03:01 So that's 0 .1 minus 1 7th, minus 5 seventh, and keep everything else.

03:13 0 .4 minus 10 minus 6.

03:27 Okay.

03:30 So then i want to keep the first row because i finish reducing the first row...

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