Linear Independence: Definition & Examples

Instructor: Sharon Linde
Do you know what linear independence is and when it is used? This lesson gives a definition of linear independence, how to test for it, and also goes over some examples of the testing process.

Definition of Linear Independence

Gus is a physics student who has a paper due soon on linear independence. How the heck is he going to tackle this? In his research, he obtained the following three equations from some experiments in a physics laboratory:

x + y + z = 0

2x - y + z = 0

3x + 2y - z = 0

If these three equations are independent of each other, Gus will be able to use them to get his research paper written. If these three equations are dependent on each other, Gus could still do all that work only to find out his starting point was poorly chosen. Is there a way for Gus to know if the three equations are independent or dependent? Luckily for Gus, there is a way to check this before he does all that work: linear independence.

What Gus wants to know, and what linear independence can help him with, is if the three equations all give him new information or if at least one of the three equations can be expressed as a combination of the other two. If at least one of the equations can be described in terms of the other equations, the system is said to be linearly dependent. If there is no way to write at least one equation as a linear combination of the other equations, then the system is linearly independent.

Testing for Linear Independence

Now that we know what linear independence means, is there a way to test for it? It turns out there is a mathematical way to test for this information, using matrix determinants. A determinant is a single number obtained by multiplying and adding the elements of a square matrix is a specific way. If the determinant of the matrix corresponding to the three equations Gus has is zero, then his three equations are dependent and he'll have to go back and set up more experiments and take more measurements before writing his paper. On the other hand, if the determinant is anything other than zero his equations are independent and he can go ahead with the work to write up his paper without fear of that effort being wasted.

Let's look at some simple examples before looking as Gus' equations.

Simple Examples of Linear Independence Test

Suppose you have the following two equations:

x + 3y = 0

2x + 6y = 0

To the trained eye, it should be obvious that the two equations are dependent on each other because the second equation is just a multiple of the first. Let's check it using determinants though.

The above equations can be written in matrix form. We do this by putting the coefficients from each type of variable in their own columns, and then we can calculate the determinant.

linear independence

The determinant of this matrix is just (1)(6) - (2)(3) = 6 - 6 = 0. Since the determinant of the equivalent matrix is equal to 0, that means the system of equations is linearly dependent.

Let's try another one before checking Gus' equations:

x - y = 0

2x + 6y = 0

Did you get that the second equation is not simply a multiple of the first equation? Let's check using the determinant. In matrix form this system looks like:

linear independence

The determinant for this matrix is (1)(6) - (2)(-1) = 6 - (-2) = 6 + 2 = 8. This determinant is not zero, and therefore this set of equations is linearly independent.

Larger Example of Linear Independence Test

Now that we know how the test works, let's see if Gus can start working on his paper or if he has more work to do in the lab. Is his set of equations linearly independent?

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