# 1.19 Linear spaces

$$\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}} \def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}} \def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)} \def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}} \def\ker{\mathbf{ker}}\def\im{\mathbf{im}}$$

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 $$\exp(2x)-x^2$$.