Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last lecture we saw how an augmented matrix
can be simplified through the process of "Gauss-Jordan elimination".
This is accomplished by performing a sequence of elementary row operations
which simplify the matrix one entry at a time
until the matrix is in "reduced row echelon form".
If the system of equations has a single unique solution
the end result will be a matrix whose coordinate entries are all ones
along the main diagonal of the matrix
and zeros everywhere else.
Since a system of linear equations in three variables
is graphically represented by a collection of planes
following how these planes change their orientation with each row operation
can help give us an intuitive understanding
of how the transformation to reduced row echelon form works.
So what effect do each of the three elementary row operations
swap
scale
and pivot
have on the planes?
The swap operation has no effect
since it only changes the order in which the matrix rows are listed.
Likewise, the scale operation which multiplies any row by a non-zero constant
does not affect the planes
since any multiple of a matrix row has the same graph as the original row.
Only the pivot operation changes the planes
since adding a multiple of one row
to another row
changes the solutions and graph of the equation which that row represents.
So how exactly does the orientation of the planes change
when a pivot operation is performed?
In the lecture entitled "A Geometrical View of the Elimination Method"
we showed that when multiples of equations for two intersecting planes are added
the resulting plane will include the same intersection line as the original two planes.
Depending upon the value of the multiplier
the resulting plane can be oriented in any way around the line of intersection.
If a multiplier is chosen so that a variable is eliminated
then the resulting plane will be parallel to that variable's axis.
For instance, if the variable z is eliminated
then the resulting plane will be parallel to the z-axis.
This is exactly what happens during a pivot operation
except that one of the original planes is replaced by the resulting plane.
So let's reduce a matrix to reduced row echelon form using Gauss-Jordan elimination
and follow how the planes change their orientation with each row operation
We will start with a system of three linear equations in three variables
represented as an augmented matrix.
Since this matrix represents a linear system in three variables
each matrix row corresponds to a plane in three-dimensional space.
We will start by changing the first row's leading entry to a "1"
using a scale operation to multiply row one by 1/4.
Since a scale operation does not affect the matrix row's graph
this operation has no effect on the planes.
Next, we change the entry below the "1" to a "0".
We do this by using a pivot operation to add negative 4 times row one
to row two.
Since the pivot operation adds a multiple of row one
to row two
it changes row two
and causes plane two to rotate around its line of intersection with plane one.
And since the x-coefficient in row two has changed to "0"
plane two has become parallel to the x-axis.
Continuing to work from the top down
we change the bottom entry in column one to "0"
using another pivot operation to add negative 4 times row one
to row three.
Since this pivot operation added a multiple of row one
to row three
this change to row three
causes plane three to rotate around its line of intersection with plane one.
And since the x-coefficient in row three has changed to "0"
plane three has become parallel to the x-axis.
Now we move to the second column
changing the second row's leading entry to a "1"
using a scale operation to multiply row two by 1/16.
Once again, a scale operation has no effect on the planes.
The entry below that "1" is then zeroed
using another pivot operation adding negative 9 times row two
to row three.
Since this pivot operation added a multiple of row two
to row three
the change to row three
causes plane three to rotate around its line of intersection with plane two.
And since the y-coefficient in row three has changed to "0"
plane three has become parallel to the y-axis.
Since plane three is now parallel to the y-axis as well as the x-axis
it is perpendicular to the z-axis and horizontal.
Moving on to column three
we then change the third row's leading entry to a "1"
using a scale operation to multiply row three by 4/37.
And once again, this has no effect on the planes.
Now we work from bottom to top
changing the entry above the one in column three to a "0"
using another pivot operation to add 1/4 times row three
to row two.
Since this pivot operation added a multiple of row three
to row two
the change to row two
causes plane two to rotate around its line of intersection with plane three.
And since the z-coefficient in row two has changed to "0"
plane two has become parallel to the z-axis and is now vertical.
Since plane two is now parallel to both the z and x axes
it is perpendicular to the y-axis.
Moving up to the top entry in column three, we zero that entry
with another pivot operation, adding 1/4 times row three
to row one.
Since this pivot added a multiple of row three
to row one
the change to row one
causes plane one to rotate around its line of intersection with plane three.
And since the z-coefficient in row one has changed to "0"
plane one has become parallel to the z-axis and is now vertical.
To finish the row reduction process
we move to column two and zero the top entry
using a pivot operation to add 1 times row two
to row one.
Since this pivot added row two
to row one
plane one rotates around its line of intersection with plane two.
And since the y-coefficient in row one has changed to "0"
plane one has become parallel to the y-axis.
And since plane one is now parallel to both the y and z axes
it is perpendicular to the x-axis.
Since plane one is now perpendicular to the x-axis
plane two is perpendicular to the y-axis
and plane three is perpendicular to the z-axis
the equations for those planes allow us directly read off the values
for the variables x, y and z.
There are typically many different ways that an augmented matrix
can be reduced to reduced row echelon form using elementary row operations.
Depending on the particular sequence of row operations used
the intermediate orientations of the planes during the process can be different.
However, once the system is in reduced row echelon form
the end result will always be the same.
Regardless of the sequence of row operations used to get to reduced row echelon form
only the pivot operations change the orientations of the planes
each pivot operation causing a single plane to rotate around its line of intersection
with the other plane used in that operation.
If the system has a single unique solution
each plane will end up oriented perpendicular to one of the three axes
allowing the x, y and z values of the solution to be read directly from the matrix.
In the next lectures
we will see what it means for a system of linear equations to be "dependent"
and what happens to dependent equations and planes
during the process of Gauss-Jordan elimination.