Linear dependence and independence; linear combinations

[member 392791]

I cannot visualize the geometry for either of these ideas. Is it the case that two vectors can be linearly independent or dependent of each other? In which case, what is the dependency or independency based on? What are these two vectors independent or dependent of with respect to each other?

 

[Erland]

We don't say that sets of vectors are linearly dependent or independent of something, just that they are linearly dependent or independent. These are properties of sets of vectors.

Consider a plane through the origin in 3-space. Take two non-parallell vectors in this plane. As always, we choose the origin as the starting point of the vectors.
If you take a linear combination of these two vectors, you will realize, by visualization, that the resulting vector must also lie in the plane. You cannot come outside the plane by taking linear combinations of these two vectors. It also true that every vector in this plane can be written as a linear combination of these two vectors. This can be seen by forming a coordinate system for this plane based upon these two vectors, in the same way as the ordinary coordinate system is based upon the standard basis vectors, but here, the axes need not be perpendicular and the scales on the axes not the same. (If you have a good textbook, you should have a figure of this somnewhere.)

Now, take these two vectors and a third vector in the plane. One of there three vectors (in this case we can choose the third one) can then be written as a linear combination of the other two. This means that these three vectors are linearly dependent. (We can define linear dependence by this: a set of vectors are linearly dependent if one of them can be written as a linear combination of the others. Otherwise they are linearly independent. This is not the most common definition of linear dependence/independence, but it is equivalent to it, which is certainly proved in your textbook.) On the other hand, if we take a third vector outside the plane, then it cannot be written as a linear combination of the first two, and actually, none of the three can be written as a linear combination of the other two, so they are linearly independent.

Geometrically, three vectors in 3-space are linearly dependent if and only if they lie in a common plane through the origin.

You should also convince yourself that two vectors in 2- or 3-space are linearly dependent if and only if they lie on a common line through the origin, that is, that they are parallell.

 

[HallsofIvy]

Two vectors are "independent" if and only if one is not a multiple of the other- they do not both lie along a single line through the origin.

Three vectors are "independent" if and only if they do not all lie in the same plane.

In general, n vectors are "independent" if and only if they do not all lie in the same n-1 dimensional subspace.