### A Definition

A subspace is a subset of a vector space that has the same structure, in other words it is also a vector space that uses the same definitions for addition and scalar multiplication. Using this definition we can tell if a set W is a subspace of a vector space V by first checking the W is a subset of V and then verifying all of the vector space axioms using W.

However, because W is a subset of a vector space we can take certain shortcuts instead of having to verify all of the vector space axioms. There are actually three attributes of W, in addition to it being a subset of V, that we must verify. These are:

- W has a zero. It can be proven that this zero will be the same as the zero of V.
- W is closed under addition. This means that given an x and y in W, x+y is in W
- W is closed under scalar multiplication. This means that given an x in W, and c, a member of the field that V is over, c*x is in W

### Subspaces of R^{2}

The most common example of a vector space is probably R^{2} which is defined as all tuples (x, y) such that x is a real number and y is a real number. This is isomorphic to the 2-d coordinate plane. The first subspaces to look at are known as the trivial subspaces and these are simply {0}, and R^{2} itself. The other subspaces of R^{2} are composed of every line that passes through 0. It is simple to show that any line through the origin will form a subspace of the total plane. Call the subset formed by the line W and let it be all elements of R^{2} of the form

W = {(x, y): y = m*x}, m ∈ R

This means W = (x, y) wherever y = m*x, and m is a constant in R; m will be the slope of the line generated. Now, because 0=m*0 we are guaranteed that the zero is included. To prove that addition is closed, consider two elements

A = (x1, y1) = (x1, m*x1)

B = (x2, y2) = (x2, m*x2)

If we then add these two generic elements we get

A + B = (x1+x2, y1+y2) = (x1+x2, m*x1+m*x2) = (x1+x2, m*(x1+x2)) = (u, m*u)

By substituting u = (x1+x2) at the end, we can easily see that this will be a member of W. To prove it is closed under addition consider

A = (x, y), c ∈ R

c*A = (c*x, c*y) = (c*x, m*c*x) = (u, m*u)

By substituting u = c*x, we see that c*A will always be a member of W if A is. Thus all three of our requirements are satisfied.

### The Dimensions of Subspaces

One of the most important attributes of a vector space is its dimension. Because of this it is useful to look at how the dimensions of a vector space and its subspaces compare. There are two important results to remember. The first is that if V is a finite-dimensional vector space, and W is a subspace of V, then dim(W) ≤ dim(V). Furthermore it can be shown that if dim(W) = dim(V), then W = V. This means there is only one subspace of any vector space V with the same dimension as V, and that is V itself.

Looking back at our example of R^{2} we have three classes of subspaces. Because R^{2} is 2 dimensional we should find subspaces with 0, 1, and 2 dimensions. The first is our two-dimensional subspace which is R^{2} itself; this matches the earlier theorem. Our second subspace is {0} and it is zero-dimensional. Our final class of subspaces are our lines, each of which is one-dimensional. As expected, no subspace of R^{2}exists with any dimension higher than 2.

### Subspaces and Linear Transformations

Most of the useful things you can do with vector spaces have to deal with the use of linear transformations. Linear transformations have some very important attributes of their behavior that are related to subspaces. Two attributes of a linear transformation, T, are its range, denoted R(T), and its null space, denoted N(T). These are defined as

T: V->W

N(T) = {x ∈ V: T(x) = 0}

R(T) = {T(x): x ∈ V}

The first line says that T is a linear transformation from V to W, both of which are vector spaces. The second line means that the null space of T is all elements of x in V such that T(x) is zero. The third line means the range of T is all elements of the form T(x) where x is in V; all of these will be members of W.

The important thing to remember is that N(T) will always be a subspace of V, and R(T) will always be a subspace of W. To prove that N(T) is a subspace of V, it is necessary to show that

A, B ∈ N(T), c ∈ F

0 ∈ N(T)

A+B ∈ N(T)

c*A ∈ N(T)

We can do this by using the attributes of linear transformation.

T(0) = 0

T(A+B) = T(A)+T(B) = 0+0 = 0 ∴ A+B ∈ N(T)

T(c*A) = c*T(A) = c*0 = 0 ∴ c*A ∈ N(T)

A similar proof is used to show that R(T) is a subspace of W. This is a basic theorem used in linear algebra and forms the basis for many more complex theorems.

### Conclusions

Subspaces are a fundamental entity used extensively througout linear algebra. Once you get used to them, they can be a powerful tool in proving relations and aspects of linear transformations, and matrix manipulations.