Matrix - Vector Equations

A system of linear equations can always be expressed in a matrix form. For example, the system

is equivalent to the matrix equation

Algebraically, both of these express the same thing.

The matrix version of the equation has its own geometric interpretation. We think of a function which is defined for vectors by the following equation:

For each vector you substitute into this function, you get another vector out. The problem is to find an input vector which produces a result equal to . This idea is illustrated in the interactive exercise below. You will choose the input vector by moving the mouse in the graph window. It will always be drawn as a red line from the origin to the point (x,y). The output vector will be drawn in a similar fashion, always shown in blue. The desired result, , will be shown in yellow. As you move the red vector, the resulting blue vector will move in response. The goal is to make the blue vector exactly match the yellow vector. Scroll down to try the activity. Follow the numbered instructions below. At the first step, you can use the numbers already shown in the colored boxes, or you can change those to define your own equation.

Did you succeed in getting the blue vector to exactly match the yellow one? If not, you might have better luck if you use the zoom in and recenter buttons to see the vectors at higher magnification.

The activity allows you to change the entries in the matrix and in the target (yellow) vector. It is interesting to see how the matrix interpretation looks for systems which are inconsistent (no solution) or underdetermined (infinitely many solutions). To see examples of these, redefine the matrix and vector entries in the activity above. For an inconsistent system, make the second matrix column a multiple of the first, but make the vector on the other side of the equal sign something that is not a multiple of the matrix columns. That might look like this:

For an underdetermined system, make both matrix columns and the third column all multiples of each other. Here is an example of that:

  Did you find that no matter what you did to the red vector, the blue vector never moved off of one line? This is always the case for a 2 by 2 matrix equation, if the columns of the matrix are multiples of one another. If the target vector (the yellow one) is on this same line, there will be an infinite number of positions for the red vector all of which make the blue vector exactly match the yellow vector. On the other hand, if the blue vector is always confined to a single line, and the yellow vector is not aligned with this line, then there is no way to make the blue vector agree with the yellow one. In other words, there is no solution to the system.

This page concerns the matrix-vector equation view of a linear system. Use the links below to see the simultaneous linear equations view or the vector equation view.