docs: update linking

docs/mathematics/differential-geometry/differential-manifolds.md

@@ -1,6 +1,6 @@
 # Differential manifolds
 
-In the following sections of differential geometry we make use of the Einstein summation convention introduced in [vector analysis](/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates/) and $\mathbb{K} = \mathbb{R}$ or $\mathbb{K} = \mathbb{C}.$
+In the following sections of differential geometry we make use of the Einstein summation convention and $\mathbb{K} = \mathbb{R}$ or $\mathbb{K} = \mathbb{C}$.
 
 ## Definition
 
@@ -38,4 +38,4 @@ The last axiom ensures that any chart is tacitly assumed to be already contained
 
 To clarify the definitions, a passive transformation corresponds only to a descriptive transformation. Whereas an active transformation corresponds to a transformation on the manifold $M$.  
 
-A passive transformation may also be given directly by $\phi_\beta \circ \phi_\alpha: x \mapsto y$ since $\psi = \mathrm{id}$ in this case. Note that the definitions could also have been given by the inverse as the transformations are all diffeomorphisms. 
+A passive transformation may also be given directly by $\phi_\beta \circ \phi_\alpha: x \mapsto y$ since $\psi = \mathrm{id}$ in this case. Note that the definitions could also have been given by the inverse as the transformations are all diffeomorphisms. 

docs/mathematics/linear-algebra/dual-vector-spaces.md

@@ -1,6 +1,6 @@
 # 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 $\mathbb{K} = \mathbb{R} \lor\mathbb{K} = \mathbb{C}$. 
+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 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 $\lambda, \mu \in \mathbb{K}$ we have
 >
@@ -54,4 +54,4 @@ From theorem 1 it follows that for each covector basis $\{\mathbf{\hat e}^i\}$ o
 
 ??? note "*Proof*:"
 
-    Will be added later.
+    Will be added later.

docs/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.md#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.
 
@@ -110,7 +110,7 @@ The complex conjugate of an eigenvector of $A$ is also an eigenvector of $A$ wit
 
 ??? note "*Proof*:"
 
-    Let $A$ be a $n \times n$ matrix and let $\lambda_1, \dots, \lambda_n \in \mathbb{K}$ be the eigenvalues of $A$. It follows from the [fundamental theorem of algebra](../../number-theory/complex-numbers/#roots-of-polynomials) that 
+    Let $A$ be a $n \times n$ matrix and let $\lambda_1, \dots, \lambda_n \in \mathbb{K}$ be the eigenvalues of $A$. It follows from the [fundamental theorem of algebra](../number-theory/complex-numbers.md#roots-of-polynomials) that 
 
     $$
         \det (A - \lambda I) = (\lambda_1 - \lambda)(\lambda_2 - \lambda) \cdots (\lambda_n - \lambda),
@@ -182,7 +182,7 @@ for $k \in \mathbb{K}$.
 
 ### Hermitian case
 
-The following section is for the special case that a matrix is [Hermitian](../matrices/matrix-arithmatic/#hermitian-matrix). 
+The following section is for the special case that a matrix is [Hermitian](matrices/matrix-arithmetic.md#hermitian-matrix). 
 
 > *Theorem 7*: the eigenvalues of a Hermitian matrix are real.
 
@@ -264,4 +264,4 @@ The factorization $A = U T U^H$ is often referred to as the *Schur decomposition
 
 ??? note "*Proof*:"
 
-    Will be added later.
+    Will be added later.

docs/mathematics/linear-algebra/orthogonality.md

@@ -46,7 +46,7 @@ for all $\mathbf{v} \in S$, and hence $\mathbf{u}_1 + \mathbf{u}_2 \in S^\perp$.
 
 ### Fundamental subspaces
 
-Let $V$ be an Euclidean inner product space $V = \mathbb{R}^n$ with its inner product defined by the [scalar product](../inner-product-spaces/#euclidean-inner-product-spaces). With this definition of the inner product on $V$ the following theorem may be posed.
+Let $V$ be an Euclidean inner product space $V = \mathbb{R}^n$ with its inner product defined by the [scalar product](inner-product-spaces.md#euclidean-inner-product-spaces). With this definition of the inner product on $V$ the following theorem may be posed.
 
 > *Theorem 1*: let $A$ be an $m \times n$ matrix, then
 >
@@ -118,7 +118,7 @@ Known as the fundamental theorem of linear algebra. Which can be used to prove t
         S^\perp = R(X^T)^\perp = N(X),
     $$
 
-    from the [rank nullity theorem](../vector-spaces/#rank-and-nullity) it follows that
+    from the [rank nullity theorem](vector-spaces.md#rank-and-nullity) it follows that
 
     $$
         \dim S^\perp = \dim N(X) = n - r.
@@ -464,4 +464,4 @@ $$
     \mathbf{\hat x} = (A^T A)^{-1} A^T \mathbf{b},
 $$
 
-is the unique solution of the normal equations for $A$ nonsingular and consequently, the unique least squares solution of the system $A \mathbf{x} = \mathbf{b}$. 
+is the unique solution of the normal equations for $A$ nonsingular and consequently, the unique least squares solution of the system $A \mathbf{x} = \mathbf{b}$. 

docs/mathematics/linear-algebra/tensors/tensor-formalism.md

@@ -1,6 +1,6 @@
 # Tensor formalism
 
-We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n$ and a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}.$ 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 $\mathbb{K} = \mathbb{R} \lor\mathbb{K} = \mathbb{C}.$
+We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n$ and a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}.$ In the following sections we make use of the Einstein summation convention and $\mathbb{K} = \mathbb{R} \lor\mathbb{K} = \mathbb{C}$.
 
 ## Definition
 
@@ -239,4 +239,4 @@ with $u^j = g^{ij} u_i$.
 >   \mathbf{g}(\mathbf{e}^i) = g_{ij} \mathbf{\hat e}^j.
 > $$
 
-So far, a vector space $V$ and its associated dual space $V^*$ have been introduced as a priori independent entities. An inner product provides us with an explicit mechanism to construct a bijective linear mapping associated with each vector by virtue of the metric. 
+So far, a vector space $V$ and its associated dual space $V^*$ have been introduced as a priori independent entities. An inner product provides us with an explicit mechanism to construct a bijective linear mapping associated with each vector by virtue of the metric.