From 9457c2fa5ced296787aa173ca2e8d2c173b7fe7f Mon Sep 17 00:00:00 2001 From: Luc Date: Thu, 9 May 2024 14:19:40 +0200 Subject: [PATCH] Added dual vector spaces and layout for tensors. --- config/en/mkdocs.yaml | 5 ++ .../linear-algebra/dual-vector-spaces.md | 57 +++++++++++++++++++ .../mathematics/linear-algebra/eigenspaces.md | 2 +- .../linear-algebra/linear-transformations.md | 1 + .../tensors/tensor-formalism.md | 0 .../tensors/tensor-symmetries.md | 0 .../linear-algebra/tensors/volume-forms.md | 0 7 files changed, 64 insertions(+), 1 deletion(-) create mode 100644 docs/en/mathematics/linear-algebra/dual-vector-spaces.md create mode 100644 docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md create mode 100644 docs/en/mathematics/linear-algebra/tensors/tensor-symmetries.md create mode 100644 docs/en/mathematics/linear-algebra/tensors/volume-forms.md diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index 1458272..39e7fb9 100755 --- a/config/en/mkdocs.yaml +++ b/config/en/mkdocs.yaml @@ -91,6 +91,11 @@ nav: - 'Inner product spaces': mathematics/linear-algebra/inner-product-spaces.md - 'Orthogonality': mathematics/linear-algebra/orthogonality.md - 'Eigenspaces': mathematics/linear-algebra/eigenspaces.md + - 'Dual vector spaces': mathematics/linear-algebra/dual-vector-spaces.md + - 'Tensors': + - 'Tensor formalism': mathematics/linear-algebra/tensors/tensor-formalism.md + - 'Tensor symmetries': mathematics/linear-algebra/tensors/tensor-symmetries.md + - 'Volume forms': mathematics/linear-algebra/tensors/volume-forms.md - 'Calculus': - 'Limits': mathematics/calculus/limits.md - 'Continuity': mathematics/calculus/continuity.md diff --git a/docs/en/mathematics/linear-algebra/dual-vector-spaces.md b/docs/en/mathematics/linear-algebra/dual-vector-spaces.md new file mode 100644 index 0000000..27977ae --- /dev/null +++ b/docs/en/mathematics/linear-algebra/dual-vector-spaces.md @@ -0,0 +1,57 @@ +# Dual vector spaces + +We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n$. In the following sections we make use of the Einstein summation convention introduced in [vector analysis](/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates/) and $\mathbb{K} = \mathbb{R} \lor\mathbb{K} = \mathbb{C}$. + +> *Definition 1*: let $\mathbf{\hat f}: V \to \mathbb{K}$ be a **covector** or **linear functional** on $V$ if for all $\mathbf{v}_{1,2} \in V$ and $\lambda, \mu \in \mathbb{K}$ +> +> $$ +> \mathbf{\hat f}(\lambda \mathbf{v}_1 + \mu \mathbf{v}_2) = \lambda \mathbf{\hat f}(\mathbf{v}_1) + \mu \mathbf{\hat f}(\mathbf{v}_2). +> $$ + +Throughout this section covectors will be denoted by hats to increase clarity. + +> *Definition 2*: let the the dual space $V^* \overset{\text{def}} = \mathscr{L}(V, \mathbb{K})$ denote the vector space of covectors on $V$. + +Each basis $\{\mathbf{e}_i\}$ of $V$ therefore induces a basis $\{\mathbf{\hat e}^i\}$ of $V^*$ by + +$$ + \mathbf{\hat e}^i(\mathbf{v}) = v^i, +$$ + +for all $\mathbf{v} = v^i \mathbf{e}_i \in V$. + +> *Theorem 1*: the dual basis $\{\mathbf{\hat e}^i\}$ of $V^*$ is uniquely determined by +> +> $$ +> \mathbf{\hat e}^i(\mathbf{e}_j) = \delta_j^i, +> $$ +> +> for each basis $\{\mathbf{e}_i\}$ of $V$. + +??? note "*Proof*:" + + Let $\mathbf{\hat f} = f_i \mathbf{\hat e}^i \in V^*$ and let $\mathbf{v} = v^i \mathbf{e}_i \in V$, then we have + + $$ + \mathbf{\hat f}(\mathbf{v}) = \mathbf{\hat f}(v^i \mathbf{e}_i) = \mathbf{\hat f}(\mathbf{e}_i) v^i = \mathbf{\hat f}(\mathbf{e}_i) \mathbf{\hat e}^i(\mathbf{v}) = f_i \mathbf{\hat e}^i (\mathbf{v}), + $$ + + therefore $\{\mathbf{\hat e}^i\}$ spans $V^*$. + + Suppose $\mathbf{\hat e}^i(\mathbf{e}_j) = \delta_j^i$ and $\lambda_i \mathbf{\hat e}^i = \mathbf{0} \in V^*$, then + + $$ + \lambda_i = \lambda_j \delta_i^j = \lambda_j \mathbf{\hat e}^j(\mathbf{e}_i) = (\lambda_j \mathbf{\hat e}^j)(\mathbf{e}_i) = \mathbf{0}, + $$ + + for all $i \in \mathbb{N}[i \leq n]$. Showing that $\{\mathbf{\hat e}^i\}$ is a linearly independent set. + +Obtaining a vector and consequent covector space having the same dimension $n$. + +From theorem 1 it follows that for each covector basis $\{\mathbf{\hat e}^i\}$ of $V^*$ and each $\mathbf{\hat f} \in V^*$ there exists a unique collection of numbers $\{f_i\}$ such that $\mathbf{\hat f} = f_i \mathbf{\hat e}^i$. + +> *Theorem 2*: the dual of the covector space $(V^*)^* \overset{\text{def}} = V^{**}$ is isomorphic to $V$. + +??? note "*Proof*:" + + Will be added later. \ No newline at end of file diff --git a/docs/en/mathematics/linear-algebra/eigenspaces.md b/docs/en/mathematics/linear-algebra/eigenspaces.md index 9589f20..b096b48 100644 --- a/docs/en/mathematics/linear-algebra/eigenspaces.md +++ b/docs/en/mathematics/linear-algebra/eigenspaces.md @@ -42,7 +42,7 @@ Furthermore it follows from the definition that any linear combination of eigenv A \mathbf{x} - \lambda \mathbf{x} = (A - \lambda I) \mathbf{x} = \mathbf{0}, $$ - which implies that $(A - \lambda I)$ is singular and $\det(A - \lambda I) = 0$ by [definition](..//determinants/#properties-of-determinants). + which implies that $(A - \lambda I)$ is singular and $\det(A - \lambda I) = 0$ by [definition](../determinants/#properties-of-determinants). The eigenvalues $\lambda$ may thus be determined from the **characteristic polynomial** of degree $n$ that is obtained from $\det (A - \lambda I) = 0$. In particular, the eigenvalues are the roots of this polynomial. diff --git a/docs/en/mathematics/linear-algebra/linear-transformations.md b/docs/en/mathematics/linear-algebra/linear-transformations.md index 3fb3181..cb90129 100644 --- a/docs/en/mathematics/linear-algebra/linear-transformations.md +++ b/docs/en/mathematics/linear-algebra/linear-transformations.md @@ -104,6 +104,7 @@ With these definitions the following theorem may be posed. It has therefore been established that each linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ can be represented in terms of an $m \times n$ matrix. > *Theorem*: let $E = \{\mathbf{e}_1, \dots, \mathbf{e}_n\}$ and $F = \{\mathbf{f}_1, \dots, \mathbf{f}_n\}$ be two ordered bases for a vector space $V$, and let $L: V \to V$ be a linear operator on $V$, $\dim V = n \in \mathbb{N}$. Let $S$ be the $n \times n$ transition matrix representing the change from $F$ to $E$, +> > $$ > \mathbf{e}_i = S \mathbf{f}_i, > $$ diff --git a/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md b/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md new file mode 100644 index 0000000..e69de29 diff --git a/docs/en/mathematics/linear-algebra/tensors/tensor-symmetries.md b/docs/en/mathematics/linear-algebra/tensors/tensor-symmetries.md new file mode 100644 index 0000000..e69de29 diff --git a/docs/en/mathematics/linear-algebra/tensors/volume-forms.md b/docs/en/mathematics/linear-algebra/tensors/volume-forms.md new file mode 100644 index 0000000..e69de29