# 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.

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

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

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

#### Compatibility

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

#### Identity

$$I\psi=\psi I=\psi$$, $$0\psi=\psi 0=0$$

### Example of Linear Spaces

1. $$\psi=(x_1,x_2,\cdots,x_n)$$, $$\phi=(y_1,y_2,\cdots,y_n)$$, $$\psi+\phi=(x_1+y_1,\cdots,x_n+y_n)$$, $$c\psi=(cx_1,\cdots,cx_n)$$.
2. (x1,x2,⋯), $$\sum_i^\infty |x_i|^2$$ is finite. $ℓ2$-space
3. Set of continuous functions
4. Set of functions with $$\int|\psi(x)|^2dx$$ is finite. $L2$-spaces
5. 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.

Created: 2023-06-25 Sun 02:29

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