1.2 The Algebra of Matrices (W, 0824)

Matrices from a linear system

If one works through a system of linear equations, it becomes clear that an organized attack involves arranging the system into some type of array and manipulating the rows of that array.
For example, the system

3x + 4y - 4z = 0

2x - 4y + 5z = 3

7x + y         = –3

3x       - 5z = 10

might be written, in an abbreviated form, as

[  3 4 -4 :  0 ]

[  2 -4 5 :  3 ]

[  7 1 0 : –3 ]

[ 3 0 -5 : 10 ]

This "box" is the augmented matrix of the system of equations.

It is not easy to write mathematics in Google's blogger.  The augmented matrix of the system above is better written as follows.

If we ignore the column of constants, we will create the matrix of coefficients of the system.

[  3 4 -4  ]

[  2 -4 5  ]

[  7 1 0   ]

[  3 0 -5  ]

This matrix has the same number of rows as the system has equations; it has the same number of columns as the system has variables.  Each row corresponds to an equation; each column corresponds to a variable.

An m x n matrix is a rectangular array of numbers, arranged in m rows and n columns. The numbers come from some set, minimally a ring but usually we will assume the elements (entries) of a matrix are from a given field.

Addition and scalar multiplication of matrices

Two matrices A and B, both of size m x n, may be added in the obvious way; we add corresponding entries to create A+B.  More formally, we will write A_{i,j} for the entry of A in row i, column j. Then

(A+B)_{i,j} = A_{i,j}+B_{i,j}.

We may multiply a matrix A by a number r ("scalar") by simply multiplying each entry of A by r. Thus

(rA)_{i,j} = rA_{i,j}.

Multiplication of matrices

Two matrices A and B may be multiplied in the order AB if and only if the number of columns of A is equal to the number of rows of B.  Matrix multiplication is more complicated than the previous two operations and is designed to mimic the composition of linear transformations.  (More on that later.)

The matrix equation corresponding to a linear system

A system of linear equations can be written in matrix notation as a single matrix equation. (This fact will motivate much of our study of matrices.)

References for matrices

More on matrices can be found at this Wikipedia article.  (The image of a matrix, above, is copied from that article.)

1.1 Systems of Linear Equations (W, 0824)

Linear equations

A linear equation is an equation involving a sum of terms, each term of which is either a constant or a single variable (to the first power) multiplied by a constant.

For example

3x + 4y-3=0

is a linear equation in the variables x and y.

However

3xy+4y-3=0

is not a linear equation as one term involves two variables (both x and y) and

3x^2+4y-3=0

is not a linear equation as one terms involves the variable x  multiplied by x.

Systems of linear equations

We are interested in systems of linear equatiions, that is, a finite set of equations in common variables.  For example

3x + 4y - 4z = 0

2x - 4y + 5z = 3

7x + y         = –3

3x       - 5z = 10

It is our goal then to find all solutions to the system.

A system of equations is homogenous is all the constant terms are zero.  The above system of four equations in three unknowns is not homogeneous.

Solving a system of linear equation

There are two fundamental techniques for solving a system of linear equations. We often will use both techniques in solving a single system.  The first technique is to replace one equation by adding a multiple of another equation to that one.  If done carefully, this will turn the linear system into an equivalent one which is simpler.

For example, in the system of four equations given above, we might replace the first equation by subtracting the fourth equation from the first.  This then gives an equivalent but simpler system:

       4y    + z  = 0

2x - 4y + 5z = 3

7x + y         = –3

3x       - 5z = 10

The second technique is that of "substitution"; we use one equation to solve for a variable and then remove that variable from consideration by replacing it in the other equations. For example, we might look at the third equation in the four-equation set above and note that y = –3-x.  We could then replace the occurrence of y in the other equations with the expression –3-x.

 This then gives an equivalent but simpler system:

–x  +  z  = 3

6x  + 5z  = –9

3x  – 5z  = 10

We may use these techniques recursively to simplify our system of equations until the solutions are clear.

The geometry of a system of equations

The solution set to a single linear equation in n variables is a "flat" object in an n-dimensional space such as R^n, the set of n-tuples of real numbers.  This solution set is called a hyperplane. For example, if we are using three variables, x, y and z then the solution set to the equation 

2x – 4y + 5z = 3

is plane in R^3, that is, a plane in 3-space.

The solution to a system of linear equations is then an intersection of those various hyperplanes.  For example, a system of three equations in three variables, x, y and z, would involve the intersection of three planes, most likely meeting in a single point, as shown in this drawing.

References

Wikipedia (of course) has a nice article on linear equations.  The drawing above is from that article.

In Fall 2016, I will be teaching a graduate class in linear algebra using an old edition of Linear Algebra by Hoffman and Kunze.  A pdf copy of that book is available on my Google Drive here.