an update of various things

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

@@ -12,7 +12,7 @@ so that $\mathbf{\hat e}^i = A^i_j \mathbf{\hat f}^j$.
 
 ## Transformation of tensors
 
-Recall from the section of [tensor-formalism]() that a holor depends on the chosen basis, but the corresponding tensor itself does not. This implies that holors transform in a particular way under a change of basis, which is characteristic for tensors. 
+Recall from the section of [tensor-formalism](tensor-formalism.md) that a holor depends on the chosen basis, but the corresponding tensor itself does not. This implies that holors transform in a particular way under a change of basis, which is characteristic for tensors. 
 
 > *Theorem 1*: let $\mathbf{T} \in \mathscr{T}^p_q(V)$ be a tensor with $p=q=1$ without loss of generality and $B = A^{-1}$. Then $\mathbf{T}$ may be decomposed into
 >
@@ -26,12 +26,20 @@ Recall from the section of [tensor-formalism]() that a holor depends on the chos
 > with the holors related by
 >
 > $$
->   \overline T^i_j = B^i_k A^j_l T^k_l.
+>   \overline T^i_j = B^i_k A^l_j T^k_l.
 > $$
 
 ??? note "*Proof*:"
 
-    Will be added later.
+    We have
+
+    $$
+        \begin{align*}
+            \mathbf{T} &= T^i_j B^k_i \mathbf{f}_k \otimes A^j_l \mathbf{\hat f}^l,\\
+            &= B^k_i A^j_l T^i_j \mathbf{f}_k \otimes \mathbf{\hat f}^l,\\
+            &= \overline T^i_j \mathbf{f}_i \otimes \mathbf{\hat f}^j.
+        \end{align*}
+    $$
 
 The homogeneous nature of the tensor transformation implies that a holor equation of the form $T^i_j = 0$ holds relative to any basis if it holds relative to a particular one.
 
@@ -54,7 +62,17 @@ The homogeneous nature of the tensor transformation implies that a holor equatio
 
 ??? note "*Proof*:"
 
-    Will be added later.
+    We have
+
+    $$
+        \begin{align*}
+            \bm{\mu} &= \mu_{i_1\dots i_n} A^{i_1}_{j_1} \mathbf{\hat f}^{j_1} \otimes \dots \otimes A^{i_n}_{j_n} \mathbf{\hat f}^{j_n},\\
+            &= A^{i_1}_{j_1} \cdots A^{i_n}_{j_n} \mu_{i_1\dots i_n} \mathbf{\hat f}^{j_1} \otimes \dots \otimes \mathbf{\hat f}^{j_n},\\
+            &= A^{i_1}_{j_1} \cdots A^{i_n}_{j_n} [i_1,\dots,i_n] \mathbf{\hat f}^{j_1} \otimes \dots \otimes \mathbf{\hat f}^{j_n},\\
+            &= \det(A) [j_1,\dots,j_n] \mathbf{\hat f}^{j_1} \otimes \dots \otimes \mathbf{\hat f}^{j_n},\\
+            &= \overline \mu_{i_1 \dots i_n} \mathbf{\hat f}^{i_1} \otimes \dots \otimes \mathbf{\hat f}^{i_1}.
+        \end{align*}
+    $$
 
 Then $\det(A)$ is the volume scaling factor of the transformation with $A$. So that if $\bm{\mu}(\mathbf{e}_1, \dots, \mathbf{e}_n) = 1$, then $\bm{\mu}(\mathbf{f}_1, \dots, \mathbf{f}_n) = \det(A).$
 
@@ -75,10 +93,6 @@ Then $\det(A)$ is the volume scaling factor of the transformation with $A$. So t
 >
 > then $\mu_{i_1 \dots i_n} = \overline \mu_{i_1 \dots i_n} = [i_1, \dots, i_n]$ is an invariant holor. 
 
-??? note "*Proof*:"
-
-    Will be added later.
-
 ## Transformation of Levi-Civita form
 
 > *Theorem 3*: let $\bm{\epsilon} \in \bigwedge_n(V)$ be the Levi-Civita tensor with
@@ -94,4 +108,4 @@ Then $\det(A)$ is the volume scaling factor of the transformation with $A$. So t
 
 ??? note "*Proof*:"
 
-    Will be added later.
+    Follows directly from the definition $\bm{\epsilon} = \sqrt{g} \bm{\mu}$.