Abstract Vector Spaces and Subspaces

Fields

A field F is a set with two binary operations (addition and multiplication) that satisfy the following axioms:

  1. Addition:
  1. is associative ((x+y)+z=x+(y+z) for all x,y,z\in F)
  2. is commutative (x+y=y+x for all x,y\in F)
  3. Has an identity element 0 such that x+0=0+x=x for all x\in F.
  4. Has an inverse -x for each x so that x+(-x)=0.
  1. Multiplication:
  1. is associative ((xy)z=x(yz) for all x,y,z\in F.)
  2. is commutative (xy=yx for all x,y\in F.)
  3. has an identity element 1 such that 1x=x for all x\in F.
  4. has an inverse x^{-1} for each nonzero element such that xx^{-1}=1.

The two operations are connected by the distributive law: for all x,y,z\in F:

x(y+z) = xy+xz

One can derive many of the standard algebraic facts from these axioms. For example, to see that 0\cdot b = 0 for all b, consider (0+0)=0

which follows from the identity axiom of addition. Then x0 = x(0+0)=x0+x0.

Now add -x0 to both sides to get

x0=0.

Similarly you can show that (-1)x=-x.

Some examples of fields:

  1. the real numbers \mathbf{R}.
  2. the complex numbers \mathbf{C}.
  3. the rational numbers \mathbf{Q}.
  4. the two-element field \mathbf{F}_{2} which consists of \{0,1\} with operations 0+0=1+1=0, 0+1=1+0=1, and all products except 1\cdot 1=1 equal to zero.

If you’re familiar with modular arithmetic, it’s also true that \mathbf{F}_{p}, the set of integers modulo p where p is prime, forms a field.

Other more complicated examples include:

  1. the field of meromorphic functions on the complex plane (or on a general Riemann surface).

  2. the field of rational functions with complex coefficients in one variable \mathbf{C}(T) or in more variables \mathbf{C}(T_1,\ldots, T_k). Here \mathbf{C} can be replaced by any other field. In fact \mathbf{C}(T) is the field of meromorphic functions on the Riemann sphere so is a special case of (a).

  3. The field of formal Laurent series with coefficients in a field F, usually written F((T)).

Vector spaces

A vector space V over a field F is a set together with two operations:

  1. Addition of elements of V satisfying:
    1. associativity (x+y)+z=x+(y+z).
    2. commutativity x+y=y+x
    3. existence of 0 such that x+0=0+x=x
    4. existence of inverses -x for each x such that x+(-x)=0.
  2. Scalar multiplication x\to ax for x\in V and a\in F such that
    1. 1x = x
    2. a(bx)=(ab)x
    3. a(x+y)=ax+ay
    4. (a+b)x= ax+bx

So our standard examples of \mathbf{C}^{n} and \mathbf{R}^{n} are vector spaces over \mathbf{C} and \mathbf{R} respectively.

Other examples:

  1. \mathbf{F}^{n}, the set of vectors of length n with entries from \mathbf{F}, where \mathbf{F} is any field.
  2. The space of continuous real-valued functions on the unit interval.
  3. The space of differentiable real-valued functions on the unit interval.
  4. The solutions to the linear nth-order differential equation

(\partial^{n}+a_{n-1}\partial^{n-1}+\cdots +a_{0})f=0

where \partial=\frac{d}{dx} and all a_{i}\in \mathbf{C}.

General Machinery

The concepts discussed for \mathbf{R}^{n} and \mathbf{C}^{n} carry over to the case of a general abstract vector space.

A map L:V\to W between \mathbf{F}-vector spaces is linear if: L(x+y)=L(x)+L(y) and L(ax) = aL(x)

  1. A subspace W of V is a subset of V that is a vector space with the inherited operations. To verify that a subset W is a subspace it’s enough to check that it’s closed under addition and scalar multiplication and contains zero.
  2. The concepts of spanning sets and linearly independent sets carry over, using coefficients from \mathbf{F}.
  3. A vector space V is finite dimensional over F if it has a finite spanning set.
  4. A basis for V is a linearly independent spanning set. Every vector space has a basis. All bases have the same cardinality; this is called the dimension of V.
  5. If A is an n\times m matrix then it defines a linear map from \mathbf{F}^{m} to \mathbf{F}^{n} by x\mapsto Ax.
  6. The row rank, column rank, rank, and nullity of a matrix with coefficients in F have the same definitions.
  7. We have \mathrm{rank}(A)+\mathrm{nullity}(A) = m where m is the number of columns of A. This follows from the fact that the row-reduction algorithm works fine for entries from a general field. In particular our proof that all bases of a finite dimensional vector space have the same number of elements works over \mathbf{F}.
  8. A linear map L:V\to W has a matrix representation [L]_{E}^{F} for bases E of V and F of W.
  9. The change of basis rules are all the same.

If f:V\to W is any linear map, then the null space (or kernel) \mathrm{null}(f) of f is the set of vectors v\in V such that f(v)=0.

The image \mathrm{image}(f) of f is the subspace \{f(v): v\in v\}\subset W.

Proposition: If V is finite dimensional then \dim(\mathrm{null}(f))+\dim(\mathrm{image}(f))=\dim(V).

The dimension of the nullspace is also called the nullity of f, and the dimension of the image is called the rank of f. This makes sense because, if you choose a basis for V and W, and find the matrix associated to f in that basis, then these numbers are respectively the rank and nullity of that matrix.

Isomorphisms

A linear map L:V\to W for vector spaces V and W over F is called an isomorphism if it is bijective. (In this case its inverse is also linear).

If there is a bijective linear map L:V\to W, then V and W are said to be isomorphic.

Now let V be a finite dimensional vector space over a field F. Choose a basis E=\{v_1,\ldots, v_n\} for V. Define a linear map L: \mathbf{F}^{n}\to V by (a_1,\ldots, a_n) \mapsto \sum_{i}a_{i}v_{i}

This is an isomorphism.

The inverse map sends v to the vector [v]_{E} = \left[\begin{matrix} a_1 \\ \vdots \\ a_n\end{matrix}\right].

So in this sense, every finite dimensional vector space over \mathbf{F} is isomorphic to \mathbf{F}^{n}, in many different ways corresponding to the choice of different bases.