1.19 Linear spaces

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

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:

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.


  • 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,

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

(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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