diff --git a/docs/en/mathematics/linear-algebra/orthogonality.md b/docs/en/mathematics/linear-algebra/orthogonality.md
index 3d23217..936a8c2 100644
--- a/docs/en/mathematics/linear-algebra/orthogonality.md
+++ b/docs/en/mathematics/linear-algebra/orthogonality.md
@@ -205,7 +205,7 @@ Recall that the system $A \mathbf{x} = \mathbf{b}$ is consistent if and only if
 
 In working with an inner product space $V$, it is generally desirable to have a basis of mutually orthogonal unit vectors.
 
-> *Definition 3*: the set of vectors $\{\mathbf{v}_i\}_{i=1}^n$ in an inner product space $V$ is **orthogonal** if 
+> *Definition 4*: the set of vectors $\{\mathbf{v}_i\}_{i=1}^n$ in an inner product space $V$ is **orthogonal** if 
 >
 > $$
 >   \langle \mathbf{v}_i, \mathbf{v}_j \rangle = 0,
@@ -215,7 +215,7 @@ In working with an inner product space $V$, it is generally desirable to have a
 
 For example the trivial set $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ is an orthogonal set in $\mathbb{R}^3$. 
 
-> *Theorem 3*: if $\{\mathbf{v}_i\}_{i=1}^n$ is an orthogonal set of nonzero vectors in an inner product space $V$, then $\{\mathbf{v}_i\}_{i=1}^n$ are linearly independent.
+> *Theorem 4*: if $\{\mathbf{v}_i\}_{i=1}^n$ is an orthogonal set of nonzero vectors in an inner product space $V$, then $\{\mathbf{v}_i\}_{i=1}^n$ are linearly independent.
 
 ??? note "*Proof*:"
 
@@ -235,7 +235,7 @@ For example the trivial set $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ is an ort
 
 We may even go further and define a set of vectors that are orthogonal and have a length of $1$, a unit vector by definition. 
 
-> *Definition 4*: an **orthonormal** set of vectors is an orthogonal set of unit vectors.
+> *Definition 5*: an **orthonormal** set of vectors is an orthogonal set of unit vectors.
 
 For example the set $\{\mathbf{u}_i\}_{i=1}^n$ will be orthonormal if and only if
 
@@ -249,7 +249,7 @@ $$
     \delta_{ij} = \begin{cases} 1 &\text{ for } i = j, \\ 0 &\text{ for } i \neq j.\end{cases}
 $$
 
-> *Theorem 4*: let $\{\mathbf{u}_i\}_{i=1}^n$ be an orthonormal basis of an inner product space $V$. If
+> *Theorem 5*: let $\{\mathbf{u}_i\}_{i=1}^n$ be an orthonormal basis of an inner product space $V$. If
 >
 > $$
 >   \mathbf{v} = \sum_{i=1}^n c_i \mathbf{u}_i,
@@ -301,7 +301,7 @@ Implying that it is much easier to calculate the coordinates of a given vector w
         \mathbf{w} = \sum_{i=1}^n b_i \mathbf{u}_i,
     $$
 
-    by theorem 4 we have 
+    by theorem 5 we have 
 
     $$
         \langle \mathbf{v}, \mathbf{w} \rangle = \Big\langle \sum_{i=1}^n a_i \mathbf{u}_i, \mathbf{w} \Big\rangle = \sum_{i=1}^n a_i \langle \mathbf{w}, \mathbf{u}_i \rangle = \sum_{i=1}^n a_i b_i. 
@@ -335,7 +335,7 @@ Implying that it is much easier to calculate the coordinates of a given vector w
 
 ### Orthogonal matrices
 
-> *Definition 5*: an $n \times n$ matrix $Q$ is an **orthogonal matrix** if 
+> *Definition 6*: an $n \times n$ matrix $Q$ is an **orthogonal matrix** if 
 >
 > $$
 >   Q^T Q = I.
@@ -343,7 +343,7 @@ Implying that it is much easier to calculate the coordinates of a given vector w
 
 Orthogonal matrices have column vectors that form an orthonormal set in $V$, as may be posed in the following theorem.
 
-> *Theorem 5*: let $Q = (\mathbf{q}_1, \dots, \mathbf{q}_n)$ be an orthogonal matrix, then $\{\mathbf{q}_i\}_{i=1}^n$ is an orthonormal set.
+> *Theorem 6*: let $Q = (\mathbf{q}_1, \dots, \mathbf{q}_n)$ be an orthogonal matrix, then $\{\mathbf{q}_i\}_{i=1}^n$ is an orthonormal set.
 
 ??? note "*Proof*:"
 
@@ -408,7 +408,7 @@ $$
 
 We wish to find a vector $\mathbf{x} \in \mathbb{R}^n$ for which $\|\mathbf{r}(\mathbf{x})\|$ will be a minimum. A solution $\mathbf{\hat x}$ that minimizes $\|\mathbf{r}(\mathbf{x})\|$ is a *least squares solution* of the system $A \mathbf{x} = \mathbf{b}$. Do note that minimizing $\|\mathbf{r}(\mathbf{x})\|$ is equivalent to minimizing $\|\mathbf{r}(\mathbf{x})\|^2$.
 
-> *Theorem 6*: let $S$ be a subspace of $\mathbb{R}^m$. For each $b \in \mathbb{R}^m$, there exists a unique $\mathbf{p} \in S$ that suffices
+> *Theorem 7*: let $S$ be a subspace of $\mathbb{R}^m$. For each $b \in \mathbb{R}^m$, there exists a unique $\mathbf{p} \in S$ that suffices
 >
 > $$
 >   \|\mathbf{b} - \mathbf{s}\| > \|\mathbf{b} - \mathbf{p}\|,
@@ -440,7 +440,7 @@ $$
 
 Uniqueness of $\mathbf{\hat x}$ can be obtained if $A^T A$ is nonsingular which will be posed in the following theorem.
 
-> *Theorem 7*: let $A \in \mathbb{R}^{m \times n}$ be an $m \times n$ matrix with rank $n$, then $A^T A$ is nonsingular. 
+> *Theorem 8*: let $A \in \mathbb{R}^{m \times n}$ be an $m \times n$ matrix with rank $n$, then $A^T A$ is nonsingular. 
 
 ??? note "*Proof*:"