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Linear Independence In Vector Spaces

Linear Independence In Vector Spaces

Author: c o

To introduce linearly independant sets of vectors
To determine when a set is linearly independent
To introduce bases of vector spaces

This packet introduces the notion of a linearly independant set of vectors and how it relates to bases of vector spaces.

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Definition Of Linear Independence

Before beginning with this packet, you should be familiar with vector spaces and about subspaces of vector spaces. It would also be helpful to be familiar with Gaussian elimination and homogeneous linear systems.

Two Definitions Of Linear Independence

Example 1


Example 2

Example - Exploring Linear Independence

This example shows how we might determine whether or not two vectors are linearly independent, and it gives a little intuition about what linear independence means.

Exotic Examples

(Disclaimer: Not all examples are terribly exotic)

Example 1 - Linear Independence In The Space Single Variable Quadratics

Example 2 - Intersecting Planes

Two planes in three dimensional space can either intersect or they can be parallel (or, the third option, where the two planes are really the same plane, in which case they intersect at every point).  If they are not parallel, then they intersect at a line.  Likewise, three planes can then either all intersect at one line, or can all intersect at one point, or they will not all intersect.  These situations can be related to linear independence.  

Consider the following equations: 

x + 3y - 2z = 0

20x - y - z = 0 

2x + 14y - z = 0

We claim that these equations are linearly independent, that if thought of as row-vectors (1,3,-2), (20,-2,-1), (2, 14, -1) in R3 then none of them is in the span of the others.  If we graph these planes, we see the following picture.

Linearly Independent Planes

The above illustrates that the three plans intersect at just one point.  This corresponds to the fact that the above linear system has just the trivial solution when x = y = z = 0.  And when a homogeneous solution has only the trivial solution, the row vectors are linearly independent.

The next example, on the other hand, shows linearly dependent equations.  Consider the following system of equations:

-x - 3y + 2z = 0

-x - y - z = 0

-2x - 4y + z = 0

In this case, a linear dependence has been introduced, we can write one of the equations as a linear combination of the others.  This corresponds to the following picture:

Linearly Dependent Planes

Now we can see that all three planes intersect at a single line.  Any of the infinitely many points on that line is a solution to the system, so that the row vectors that correspond to the system, (-1,-3,2),(-1,-1,-1), (-2,-4,1) in R3 are not linearly independent.  In fact, we can write (-2,-4,1) = (-1,-3,2) + (-1,-1,-1).