\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}}

\def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}}

\def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}

\def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}}

\def\ker{\mathbf{ker}}\def\im{\mathbf{im}}

\)

**1. ***Linear (or vector) spaces* (over a field \(k\) – which in our case will be almost always the complex numbers) are sets that satisfy the following standard list of properties:

- There exists an (associative and commutative) addition/subtraction operations, along with a neutral element, the origin,
- …and a multiplication by scalar, again associative,
- …and distributivity holds.

* Examples*: Complex numbers is a vector space over real numbers. Real numbers are a vector space over rational numbers.

**2.** The “usual” \(n\)-dimensional space consists of row- or column-vectors of length \(n\), with the standard, component-wise operations. The shape of the matrix (row, column, staircase, rectangular,…) plays, in itself, no role.

**3.** A collection of vectors in a linear space \(V\) spans or generates a linear subspace (of all linear combinations of these vectors).

**4.** A collection of vectors \(v_1,\ldots,v_m\in V\) of a linear space is *linearly dependent*, if there is a nontrivial (i.e. with not all coefficients \(0\)) vanishing linear combination of the vectors:

\[

0=c_1v_1+\ldots+c_mv_m.

\]

**5.** A linearly *independent* set of vectors that *spans* a linear space \(V\) is called a basis of \(V\). Any vector in a linear space can be *uniquely* represented as a linear combination of the basis vectors.

The size of the basis is called the *dimension* of \(V\).

**Theorem**: the size of any basis of a linear space is the same.

If this size is finite, the space is called finite-dimensional.

*Example*:

- Vectors \(1,\i\) form a basis of \(\Comp\) as a vector space over reals.
- What is the size of the basis of \(\Real\) as a vector space over rational numbers?

**5 ½. **Linear finite-dimensional subspaces of the space of functions on the line invariant with respect to *shifts *(the linear spaces \(U\) of functions such that if \(f(z)\in U\) then \(f(z+a)\) is in \(U\) as well) can be characterized as the spaces of solutions to linear differential equations (with constant coefficients).

*Exercise*: Do the polynomials

\[

p_0=1, p_1=(x-1), p_2=(x-1)^2, p_3=(x-1)^3

\]

form a basis of the vector space of cubic polynomials (with real coefficients)? If so, express \(x^3\) in this basis.

**6.** There are several important relationships between subspaces of a space. If \(U, V\) are two such subspaces, we denote by \(U+V\) the collection of sums \(u+v, u\in U, v\in V\). It is a linear space again, as \(U\cap V\) is.

One has the following “inclusion-exclusion” equality:

\[

\dim(U+V)+\dim(U\cap V)=\dim U+\dim V.

\]

It is easy to prove…

**7.** Given pair \(V\subset U\) of linear space and its subspace, one can form the factorspace, as the collection of the equivalence classes,

\[

U/V:=\{u+V\}.

\]

**8.** Dual vector spaces

If \(V\) is a linear space, then the *dual vector space* is the space of linear functions on \(V\), that is functions

\[

f:V\to k\ \mathrm{\ such\ that\ } f(c_1 v_1+c_2v_2)=c_1f(v_1)+c_2f(v_2),\ \mathrm{\ for\ any}\ c_i\in k, v_i\in V.

\]

The dual to \(V\) space is denoted as \(V^*\). Its elements sometimes are called *covectors*.

One needs to check that this is a linear space!

**9.** If \(V\) has a basic \(e_i, i=1,\ldots,n\), then there exists a natural dual basis in \(V^*\): just the functionals \(e_i^*\) given by

\[

e_i^*(e_j)=\delta_{ij}.

\]

(This again is a theorem that needs to be proven! That amounts to showing that a) we a covector is uniquely defined by just setting its values on the basis vectors; b) that the dual basis covectors span \(V^*\), and c) that they are linearly independent.)

So, the dual space to an \(n\)-dimensional space is \(n\)-dimensional too. There are a few systems of notations aimed at differentiating vectors from covectors, for example upper- and lower- indices and bra-ket notation.

*Exercise*: Let \(V\) be the linear space of quadratic polynomials, and \(e^*_k=p(k-1), k=1,2,3\). Check that these functions are linear (as functions on polynomials). Express \(p(3)\) as a linear combination of \(e_k\)s.

Find the smallest linear subspace of the space of smooth functions that is invariant with respect to shift, and containing \(\exp(2x)-x^2\).

## No comments yet.