^^ [[Schutz - 1 - Some Basic Mathematics|1. Some Basic Mathematics]] <<[[Schutz - 1.4 - Group Theory|1.4 Group Theory ]]|[[Schutz - 1.6 - The Algebra of Square Matrices| 1.6 The Algebra of Square Matrices]] >> # Linear Algebra > **Vector Spaces** > A set $V$ is a *vector space* (over a field $\mathbb{F}$, but here we restrict $\mathbb{F} = \mathbb{R}$) if it has a binary operation called $+$ with which it is an [[Schutz - 1.4 - Group Theory#^426790|Abelian group]] and if multiplication $\cdot$ by real numbers is defined to satisfy the following axioms: > Given $\pmb{x}, \pmb{y} \in V, \, a, b \in \mathbb{R}$ > 1. $a \cdot(\pmb{x} + \pmb{y}) = (a \cdot \pmb{x}) + (b \cdot \pmb{y})$ > 2. $(a + b) \cdot \pmb{x} = (a \cdot \pmb{x}) + (b \cdot \pmb{x})$ > 3. $(ab) \cdot \pmb{x} = a \cdot (b \cdot \pmb{x})$ > 4. $1 \cdot \pmb{x} = \pmb{x}$ > > The identity element of the Abelian group $V$ is called $\pmb{0}$. ^c98ca7 > **Linear Independence** > A set of elements $\{\pmb{x}_1, \pmb{x}_2, \ldots \pmb{x}_m\} \in V$ is said to be *linearly independent* if it is impossible to find a set of real numbers $\{a_1, a_2, \ldots a_m\} \in \mathbb{R}^*$ for which: > $a_1 \pmb{x}_1 + a_2 \pmb{x}_2 + \ldots a_m \pmb{x}_m = \pmb{0}$ ^cbdeb1 > **Basis, maximal sets, and dimensions** > The set is called a *maximal linearly independent set* if including any other vector of $V$ in it would make it [[Schutz - 1.5 - Linear Algebra#^cbdeb1|linearly dependent]] > By definition, this means that any other vector in $V$ can be expressed as a linear combination of elements in a maximal set, so a maximal set forms a *basis* for $V$. > The number of vectors in a basis is the *dimension* of $V$. All bases have the same dimension (if that number is finite). ^3e9444 > **Components of a vector** > If the vectors $\{\pmb{x}_i, i = 1, \ldots, n\}$ forms a basis of an $n$-[[Schutz - 1.5 - Linear Algebra#^3e9444|dimensional]] vector space $V$, then any $\pmb{y} \in V$ can be written as: > $ \pmb{y} = \sum_{i = 1}^{n} a_i \pmb{x}_i$ > The set of real numbers $\{a_i, i = 1, \ldots n\}$ are the *components* of $\pmb{y}$ on this basis. > **Vector Subspace** > A *vector subspace* of a [[Schutz - 1.5 - Linear Algebra#^c98ca7|vector space]] $V$ is a subset of $V$ that is itself a vector space. > Any set of vector $\{\pmb{y}_i i = 1, m\}$ is said to *generate* a subspace of $V$ which is formed by all possible linear combinations of the form: > $a_1 \pmb{y}_1 + a_2 \pmb{y}_2 + \ldots a_m \pmb{y}_m$ ^176b33 > **Proper Subspace** > If the basis generating the [[Schutz - 1.5 - Linear Algebra#^176b33|vector subspace]] has $m < n$, then we say that the vector subspace is a *proper subspace* of the vector space (i.e. they are not isomorphic). That is, the dimension of the subspace is smaller than that of the original space itself, and is entirely contained within it. > **Normed Vector Space** > A *normed vector space* $V$ is a [[Schutz - 1.5 - Linear Algebra#^c98ca7|vector space]] with a mapping $n:V\rightarrow\mathbb{R}$ that satisfies the axioms: > 1. $n(\pmb{x}) \geq 0 \forall \pmb{x} \in V \quad \text{and} \quad n(\pmb{x}) = 0 \iff \pmb{x} = 0$ > 2. $n(a\pmb{x}) = |a|n(\pmb{x}) \quad \forall a \in \mathbb{R} \quad \forall \pmb{x} \in V$ > 3. $n(\pmb{x} + \pmb{y}) \leq n(\pmb{x}) + n(\pmb{y}) \quad \forall \pmb{x}, \pmb{y} \in V$ ^c484af > **Euclidean Norm** > The *Euclidean norm* is one that satisfies the *parallelogram rule*: > $[n(\pmb{x} + \pmb{y})]^2 +[n(\pmb{x} - \pmb{y})]^2 = 2[n(\pmb{x}]^2 + 2[n(\pmb{y})]^2$ ^0bb145 >**Euclidean Space** > n-dimensional *Euclidean Space* $\mathbb{E}^n$ is $\mathbb{R}^n$ equipped with the [[Schutz - 1.5 - Linear Algebra#^0bb145|Euclidean norm]] > **Inner Product** > The *inner product* is a bilinear symmetric map $\cdot:V\times V^*\rightarrow \mathbb{R}$ on a [[Schutz - 1.5 - Linear Algebra#^c484af|normed vector space]] equipped with a [[Schutz - 1.5 - Linear Algebra#^0bb145|Euclidean norm]] that takes the form: > $\pmb{x} \cdot \pmb{y} = \frac{1}{4} [n(\pmb{x} + \pmb{y})]^2 - \frac{1}{4} [n(\pmb{x} - \pmb{y})]^2$ > It has the following properties: > 1. It is *bilinear*: > $(a \pmb{x} + b \pmb{y}) \cdot \pmb{z} = a (\pmb{x} \cdot \pmb{z}) + b (\pmb{y} \cdot \pmb{z})$ > $\pmb{z} \cdot (a \pmb{x} + b \pmb{y})= a (\pmb{z} \cdot \pmb{x}) + b (\pmb{z} \cdot \pmb{y})$ > 2. It is *symmetric* > $ \pmb{x} \cdot \pmb{y} = \pmb{y} \cdot \pmb{x} $ > 3. It is *positive definite* > $\pmb{x} \cdot \pmb{x} \geq 0 \quad \text{and} \quad \pmb{x} \cdot \pmb{x} = 0 \iff \pmb{x} = \pmb{0}$ ^29961c > **Pseudo-norm** > A pseudo-norm is one which violates rules (2) and (3) of the [[Schutz - 1.5 - Linear Algebra#^c484af|norm]]. It is a non-degenerate Hermitian form.