Reduced echelon form where all variables seem to = 0

In summary, reduced echelon form is a simplified and standardized way of representing a system of linear equations. It is unique because it ensures that all leading coefficients are equal to 1 and all other entries in the same column are equal to 0. This form is useful in solving systems of equations as it allows for easier identification of variables and their coefficients. It is possible to transform any system of linear equations into reduced echelon form using elementary row operations, but the solutions may not always be unique.
  • #1
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Homework Statement


Suppose I have the augmented matrix

0 -1 0 | 0
0 -6 3 | 0
0 -1 0 | 0


Homework Equations



which equates to -y = 0 and -6y + 3z = 0.


The Attempt at a Solution



Would the solution be that x, y and z all equal 0?

Or do I need to let the missing variable, x = s and free variable z = t, so the solution is

1 0
s[ 0 ] + t [ 0.5 ]
0 1

(even though I know y = 0 and therefore the free variable must(?) be 0 ) ?




Thanks.
 
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  • #2
Sort of. Yes, definitely y=0 and z=0. And x is free. z isn't.
 

FAQ: Reduced echelon form where all variables seem to = 0

What is reduced echelon form?

Reduced echelon form is a mathematical concept used to represent a system of linear equations in a simplified and standardized way. It is often used in solving systems of equations and finding solutions to linear equations.

How is reduced echelon form different from other forms of representing linear equations?

Reduced echelon form is unique because it ensures that all the leading coefficients (the first non-zero number in each row) are equal to 1 and that all other entries in the same column are equal to 0. This allows for easier identification of the variables and their corresponding coefficients.

Why do all the variables seem to equal 0 in reduced echelon form?

In reduced echelon form, the variables are represented as columns in a matrix and have been eliminated by using a series of elementary row operations. This results in a simplified form where the variables are set to 0 to make it easier to solve the equations and find their solutions.

How is reduced echelon form useful in solving systems of equations?

Reduced echelon form allows for an organized and standardized representation of a system of linear equations. This makes it easier to identify the variables and their corresponding coefficients, and to solve the system of equations by back substitution or other methods.

Can any system of linear equations be transformed into reduced echelon form?

Yes, any system of linear equations can be transformed into reduced echelon form by using elementary row operations. However, not all systems of equations will have a unique solution, and some may have no solution or infinite solutions.

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