Professor Dave here, let’s enter the matrix.
We now understand that one of the primary goals of linear algebra is solving systems
of linear equations.
In order to do this, we will frequently be expressing these equations not as we are used
to expressing them, but instead by using something called matrix notation.
So what exactly is a matrix?
Let’s go ahead and see how these can be constructed.
Say we have the following system of equations with three variables.
They all have the same variables, they just have different coefficients, and each expression
equals a different number.
Since the variables are all the same, it is the coefficients that are relevant, so why
don’t we abbreviate these coefficients some other way?
We can set up a matrix.
A matrix is an array of numbers contained in brackets.
It has rows and columns of numbers, and we can insert a number of rows and columns that
will allow us to express the coefficients in this linear system.
The three coefficients in the first equation can make up the first row.
The three coefficients in the second equation can make up the second row.
And the three coefficients in the third equation can make up the third row.
We have just constructed a coefficient matrix.
To get a matrix that contains all of the information from the system, we need to add one more column
on the right side, which will contain the numbers on the right side of the equals sign
for each equation, and this is now called an augmented matrix, which may or may not
have a dotted line here to separate this last column from the coefficients.
So there we have it.
A three by four matrix, meaning a matrix with three rows and four columns, that contains
all of the information that was contained in the system of linear equations.
We don’t need to see the variables, because they are implied, and it doesn’t matter
whether we were calling them A, B, and C, or X, Y, and Z, or anything else you can think of.
What matters are the coefficients, and the numbers on the other side of the equations.
We should note something important here, we need all of the equations to be in the same
format for this to work.
If one of the equations had two variables on the left side and the third variable on
the other side, we would have to bring that third variable over with the others by addition
or subtraction.
Also, if one of the variables is missing from one of the equations, we have to put it back
in, but with a coefficient of zero.
This will not change the equation, but it gives us a coefficient of zero to plug into
our matrix.
In general, a system with M equations and N variables will give us an M by N coefficient
matrix, because each equation gets its own row, and each variable gets its own column.
If we then include the constant terms on the other side of each equation, we will get an
augmented matrix with dimensions M by N plus one.
Let’s practice making an augmented matrix one time.
Here are five equations, with four variables.
This doesn’t look like it’s in the right form yet, so let’s make some changes.
First, let’s bring these variables that are on the right over to the left side with
the other variables.
We have to make sure that the variables are in the same order in each equation, or that
will mess up our matrix.
Next, anywhere that one of the four variables is missing from an equation, let’s put it
back in there, but with a coefficient of zero.
Now we are all set.
Each equation has the same variables in the same order, and all the variables are present
with some coefficient, with a constant on the other side of the equals sign.
Now, to make the augmented matrix, we just take all of these numbers, and place them
where they go, equation by equation, row by row, and this resulting five by five matrix
is what we will get.
So that’s all there is to representing systems of linear equations in matrix notation.
We have a lot left to learn regarding what to do with matrices, but before we move forward,
let’s check comprehension.