Why a subspace of a vector space is useful

Solution 1:

It can help to think of these concepts geometrically.

In the context of our 3d world, subspaces might be thought of as lines or planes (through the origin).

Why do we care about subspaces? Again, a geometric picture of linear transformations (which is what we use matrices to model) helps with these ideas. A linear transformation (matrix) might leave certain lines invariant: they simply map the line to the same line, within a scaling factor. Any vector on such a line is an eigenvector, and the scale factor by which the line is magnified or shrunk is the eigenvalue.

A linear transformation (matrix) might, even when given any vector in 3d, only spit out vectors on a certain plane or line. The set of vectors spat out in this way is the image of the transformation, and you should see that the dimensionality of the image is clear from its geometric dimension: if the image is a plane, then the image has dimension 2, and so on.

For inputs to the transformation (matrix), some lines or planes might be wholly annihilated by the transformation---the transformation (matrix) forces them to zero. These lines or planes form the kernel of the transformation (matrix).

Now, something you might not be taught about are whole planes that are left invariant under a transformation, even though no individual line is kept invariant. These planes might be scaled by some factor, and so they can be thought as "eigenplanes". Rotation maps are an example of transformations that leave whole planes invariant without leaving any individual (real) line in that plane invariant.

Solution 2:

An example, among many, of the usefulness of the concept of subspaces is that it is itself a vectorspace. Hence once a vectorspace has been built, one can construct many more examples by considering its vectorspace.

Also, it gives us an easy way to check that a space is a vectorspace. Instead of having to check all the axioms, we may check that some space is a subspace of a vectorspace and then conclude that it is also a vectorspace.

Solution 3:

I'd say the the root of the fact that subspaces are important have to do a lot with linear transformations:

It is not hard to show that the nullspace (or kernel) and the image of a linear transformation are vector spaces (i.e. subspaces of the domain and the codomain, respectively).

Since these two things in large part form the basis (no pun intended) of most of linear algebra to some degree, being able to know that you needn't consider anything wild or different when a linear transformation is not injective or surjective is a useful thing.