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content/notes/math/linear_algebra.mdx

@@ -4429,9 +4429,9 @@ If $\mu = pq$, then there is a $\mathbf{v}\in M$ for which $\mathbf{w} = q\mathb
 | $\mathbb{F}[x]$-module $V_\mathrm{T}$ | $\mathbb{F}$-vector space $V$ |
 | --- | --- |
 | Scalar multiplication: $p(x)\mathbf{v}$ | Action of $p(\mathrm{T})$: $p(\mathrm{T})\mathbf{v}$ |
-| Annihilator: $\operatorname{ann}(V_\mathrm{T}) = \Set{p(x)| p(x)V_\mathrm{T} = \Set{0}}$ | Annihilator: $\operatorname{ann}(V) = \Set{p(x) | p(\mathrm{T})(V) = \Set{0}}$ |
+| Annihilator: $\operatorname{ann}(V_\mathrm{T}) = \Set{p(x) : p(x)V_\mathrm{T} = \Set{0}}$ | Annihilator: $\operatorname{ann}(V) = \Set{p(x) p(\mathrm{T})(V) = \Set{0}}$ |
 | Monic order $m(x)$ of $V_\mathrm{T}$: $\operatorname{ann}(V_\mathrm{T}) = \langle m(x) \rangle$ | Minimal polynomial of $\mathrm{T}$: $m(x)$ has the smalles degree with $m(\mathrm{T}) = 0$ |
-| Cyclic submodule of $V_\mathrm{T}$: \langle \mathbf{v}\rangle = \Set{p(x)\mathbf{v} | \deg(p(x)) < \deg(m(x))} | $\mathrm{T}$-cyclic subspace of $V$: \langle \mathbf{v}, \mathrm{T}\mathbf{v},\dots, \mathrm{T}^{m-1}(\mathbf{V})\rangle,\; m = \deg(p(x)) |
+| Cyclic submodule of $V_\mathrm{T}$: $\langle \mathbf{v}\rangle = \Set{p(x)\mathbf{v} : \deg(p(x)) < \deg(m(x))}$ | $\mathrm{T}$-cyclic subspace of $V$: $\langle \mathbf{v}, \mathrm{T}\mathbf{v},\dots, \mathrm{T}^{m-1}(\mathbf{V})\rangle,\; m = \deg(p(x))$ |
 
 Let $V$ be a finite-dimensional vector space. If $\mathrm{T}\in\mathcal{L}(V)$, we will think of $V$ not only as a vector space over a field $\mathbb{F}$, but also as a module over $\mathbb{F}[x]$ with scalar multiplication defined by
 
@@ -4548,7 +4548,7 @@ $$
 Hence, the set
 
 $$
-  B = \Set{\mathbf{v}, x\mathbf{v},\dots,x^{n-1}\mathbf{v}} = \Set{\mathbf{v}\mathrm{T}\mathbf{v},\dots,\mathrm{T}^{n-1}\mathbf{v}}
+  B = \Set{\mathbf{v}, x\mathbf{v},\dots,x^{n-1}\mathbf{v}} = \Set{\mathbf{v},\mathrm{T}\mathbf{v},\dots,\mathrm{T}^{n-1}\mathbf{v}}
 $$
 
 spans the *vector space* $\langle\mathbf{v}\rangle$. To see that $B$ is a basis for $\langle\mathbf{v}\rangle$, note that any linear combination of the vectors in $B$ has the form $r(x)\mathbf{v}$ for $\deg(r(x)) < n$ and so is equal to $0$ if and only if $r(x) = 0$. Thus, $B$ is an ordered basis for $\langle\mathbf{v}\rangle$.
@@ -4604,7 +4604,7 @@ $$
 </MathBox>
 
 <MathBox title='Primary cyclic decomposition theorem for vector spaces' boxType='theorem'>
-Let $V$ be a finite-dimensional vector space and let $\mathrm{T}\in\mathcal{L}(V)$ have the minimal polynomial $m_\mathrm{T} (x) = \prod_{i=1}^n p_i^{e_i} (x)$, where the polynomials $PAGES_DIR_ALIAS(x)$ are distinct monic primes.
+Let $V$ be a finite-dimensional vector space and let $\mathrm{T}\in\mathcal{L}(V)$ have the minimal polynomial $m_\mathrm{T} (x) = \prod_{i=1}^n p_i^{e_i} (x)$, where the polynomials $p_i(x)$ are distinct monic primes.
 1. **Primary decomposition:** The $\mathbb{F}[x]$-module $V_\mathrm{T}$ is the direct sum $V_\mathrm{T} = \bigoplus_{i=1}^n V_{p_i}$, where
 $$
   V_{p_i} = \frac{m_\mathrm{T}(x)}{p_i^{e_i}(x)}V = \Set{\mathbf{v}\in V | p_i^{e_i}(\mathrm{T})(\mathbf{v}) = 0}
@@ -4624,7 +4624,7 @@ $$
   B_{i,j} = (\mathbf{v}_{i,j}, \mathrm{T}\mathbf{v}_{i,j},\dots,\mathrm{T}^{d_{i,j}-1}\mathbf{v}_{i,j})
 $$
 and $\dim(\langle\mathbf{v}_{i,j}\rangle) = \deg(p_i^{e_{i,j}})$. Hence, $\dim(V_{p_i}) = \sum_{j=1}^{k_i} \deg(p_i^{e_{i,j}})$. The basis $R = \bigcup_{i,j} B_{i,j}$ is called the *elementary divisor basis* for $V_\mathrm{T}$. 
-<MathBox>
+</MathBox>
 
 Recall that if $V = A \oplus B$ and if both $A$ and $B$ are $\mathrm{T}$-invariant subspaces of $V$, the pair $(A,B)$ *reduces* $\mathrm{T}$. In module terms, the pair $(A,B)$ reduces $\mathrm{T}$ if $A$ and $B$ are submodules of $V_\mathrm{T}$ and $V_\mathrm{T} = A_\mathrm{T} \oplus B_\mathrm{T}$.