MATH 240

Wed. October 9th, 2019


Vector Spaces and Subspaces

Definition. A vector space is a non-empty set VV of objects called vectors on which are defined two operations called addition and multiplication by scalars subject to ten axioms.

For all vectors u,vu,v and scalars c,dc,d:

  1. If u,vu,v are in VV, then u+vu+v is also in VV.
  2. u+v=v+uu+v=v+u.
  3. u+(v+w)=(u+v)+wu+(v+w)=(u+v)+w.
  4. There is a zero vector 00 in VV such that u+0=uu+0=u.
  5. For each uu in VV, there is a vector u-u in VV such that u+(u)=0u+(-u)=0.
  6. If uu is in VV and cc is a scalar, cucu is also in VV.
  7. c(u+v)=cu+cvc(u+v)=cu+cv.
  8. (c+d)u=cu+du(c+d)u=cu+du.
  9. c(du)=(cd)uc(du)=(cd)u.
  10. 1u=u1u=u.

Note: The scalars we will be working with are almost always real numbers, but they could also be complex numbers.

Also note that by this definition, the real numbers are considered to be a vector space, as are the complex numbers.


Types of Vector Spaces

Finite-Dimensional Vector Spaces

We mostly deal with finite-dimensional vector spaces – they look like Rn\R^n, where nn is a positive nonzero integer. This means that the size of any basis in these vector spaces will be finite.

Infinite-Dimensional Vector Spaces

There are also infinite-dimensional vector spaces – these can’t be represented in an Rn\R^n-form. For example, the set of all continuous functions on the interval [0,1][0,1], denoted by C[0,1]C[0,1], can be considered a vector space with infinite dimensions. It fulfills the ten axioms:

The set of all functions on R\R which are infinitely differentiable, denoted as C(R)C^\infin(\R), is also an infinite-dimensional vector space.

A third example of an infinite-dimensional vector space is the set of all sequences that are absolutely convergent.

Applying Linear Transformations

The reason it’s important to talk about linear transformations, even though so far we’ve just been able to look at matrices to do computations, is because there is no good analogue to matrices when dealing with infinite-dimensional vector spaces.

For example, integration is a linear transformation: T:C[0,1]R, T(f)=01f(x)dx.T:C[0,1]\rightarrow\R, \ T(f)=\int_0^1f(x)dx.
The derivative is also a linear transformation: T:C(R)C(R), T(f)=ddxf(x).T:C^\infin(\R)\rightarrow C^\infin(\R), \ T(f)=\frac{d}{dx}f(x).

Many important objects in math turn out to be vectors, and many important operations in math turn out to be linear transformations!