1.19 Linear spaces

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

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