# Linear Spaces

## Definition

A linear (vector) space is a set of elements (vectors) with an operation of vector addition and scalar multiplication.

Let \(\psi,\phi,\chi\) be vectors in \(\mathcal{V}\). And \(c,d\) are scalars.

### Addition Rules

Their sum is a vector in \(\mathcal{V}\), \((\psi+\phi)\in \mathcal{V}\)

#### Commutativity

\(\psi+\phi=\phi+\psi\)

#### Associativity

\((\psi+\phi)+\chi=\psi+(\phi+\chi)\)

#### Zero vector

\(0 + \psi = \psi\)

#### Inverse vector

\(\psi + (-\psi) = 0\)

### Multiplication Rules

\(c\psi\in \mathcal{V}\), \((c\psi+d\phi)\in \mathcal{V}\)

#### Distributivity over addition

\(c(\psi+\phi)=c\psi+c\phi\)

#### Compatibility

\(c(d\psi)=(cd)\psi\)

#### Identity

\(I\psi=\psi I=\psi\), \(0\psi=\psi 0=0\)

### Example of Linear Spaces

- \(\psi=(x_1,x_2,\cdots,x_n)\), \(\phi=(y_1,y_2,\cdots,y_n)\), \(\psi+\phi=(x_1+y_1,\cdots,x_n+y_n)\), \(c\psi=(cx_1,\cdots,cx_n)\).
- (x
_{1,x}_{2,⋯}), \(\sum_i^\infty |x_i|^2\) is finite. $ℓ^{2}$-space - Set of continuous functions
- Set of functions with \(\int|\psi(x)|^2dx\) is finite. $L
^{2}$-spaces - Hilbert Space

## Quantum States

### State Vector

\(|\psi\rangle \in \mathcal{E}\), \(\mathcal{E}\) is the state space which is a subset of \(L^2\) which is a subset of a Hilbert space.

### Scalar Product

\((\varphi_i,\varphi_j)=\delta_{ij}\) implies that the vectors are orthonormal.

The expansion coefficients of \(\psi\) can be found with the scalar product of the base vector with \(\psi\).

### Linear Independence Examples

#### Example 1

Independent.

\(f(x) = x, g(x) = x^2, h(x) = \exp(2x)\)

#### Example 2

Dependent.

\(f(x) = x, g(x) = 5x, h(x) = x^2\)

#### Example 3

Dependent.

\(f(x) = \cos x, g(x) = \exp ix, h(x) = \sin x\)

## Wronskian Functional

The Wronskian can be used to determine linear independence: \(\begin{vmatrix}f(x) & g(x) & h(x) & \cdots \\f'(x) & g'(x) & h'(x) & \cdots \\ f''(x) & g''(x) & h''(x) & \cdots\\\cdots & \cdots &\cdots & \cdots\end{vmatrix}\). Note that the determinant entries are \(N\times N\), where \(N\) is the number of functions you are comparing for linear independence. Hence you would only go up to \(g'\) if you only had 2 functions.