Fields, vector spaces, subspaces, linear operators, range space, null space
A field \((F,+,·)\) is a set with two commutative associative operations with \(0\) and\(1\), additive and multiplicative inverses and subject to the distributive law.
Examples (Common fields)
\((\Real,+,·)\) is a field. Rational, Complex numbers. Matrices (when they are fields?)
Vector spaces – modules over a field. (Real definition: abelian group with an action of \(\Field\) and several natural axioms.)
Fun(\(A,\Field\)), for any \(A\) – vector space over \(\Field\). (For example, sequences…)
The standard \(n\)-dimensional vector space over \(\Field\).
Kronecker delta’s. Finite sums of delta’s.
What if the set \(A\) is infinite? Finite sums are not enough: many more functions.
Subclasses of functions: continuous, integrable,…
Linear subspace (of a linear space)- subset that is a linear space (over the same field, of course). Easiest way to fix a linear subspace: by a linear equation.
Set of functions vanishing on \(B\subset A\) (fixing other values, non-zeros, would it work?).
All but finite values vanish. (What about vanishing ni at least one point?)
Functions with finite number of discontinuities?
Also form a linear space! Examples.
Nice criterion of linear independence: if \(e_1,\ldots,e_n\in V\) is a set of vectors, and \(f_1,\ldots,f_n\in V^*\) a set of linear functions, and
\vdots &\ddots &\vdots \\
f_n(e_1) & \ldots&f_n(e_n)\\
then both sets, \(e\)’s and \(f\)’s are linear independent.
Bases (a set of vectors is a basis if it is linearly independent and spans the space).
In particular, there is a unique representation of any vector in a basis.
If there exists a finite basis, the space is finite-dimensional.
Representing a vector in a basis involves a duel basis!
Dimension (the size of a basis) is well-defined.
Theorem. Any linearly independent set can be completed to a basis.
Change of basis:
x=\sum_k a(k) e^k\times \sum_l f_l f^l=\sum_l (\sum_k P(k,l)a(k)) f^l.