\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}

\def\Field{\mathbb{F}}

\)

#### 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.)

**Examples**

**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.

##### Examples:

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?

#### Linear functions

Also form a linear space! Examples.

#### Linear independence.\

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

\[

\left|

\begin{array}{ccc}

f_1(e_1)&\ldots &f_1(e_n)\\

\vdots &\ddots &\vdots \\

f_n(e_1) & \ldots&f_n(e_n)\\

\end{array}

\right|\neq 0,

\]

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.

\]

Here \(P(k,l)=f_l(e^k)\).

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