LU decomposition typo & suggestion

semester1/linalg/linalg.typ

@@ -175,7 +175,7 @@ Basically $A x = b$.
 
 == LU Decomposition
 
-Run REF on $A in RR^(m times n)$ to generate $L in RR^(m times n)$ and track coefficients in $U in RR^(n times n)$.
+Run REF on $A in RR^(m times n)$ to generate $U in RR^(m times n)$ and track coefficients in $L in RR^(n times n)$, where $L$ contains $1$'s on the diagonal and the opposite values of the operations performed on the corresponding rows.
 
 == Permutation Matrices
 
@@ -518,4 +518,4 @@ Any matrix $A$ can be factored as $A = U Sigma V^T$.
 - $A^T A = U Lambda_1 U^T$. Here we have that $Lambda_1 = Sigma^T Sigma$. $Sigma = "diag"(sigma_1, ..., sigma_k) "s.t." k = min(n,m)$
 - $A A^T = V Lambda_2 V^T$. Here we have that $Lambda_2 = Sigma Sigma^T$. $Sigma = "diag"(sigma_1, ..., sigma_k) "s.t." k = min(n,m)$
 - $sigma_i = sqrt(lambda_i)$.
-- For both: $Sigma$ is constructed s.t. $sigma_1 >= ... >= sigma_k >= 0$. Rank: number of non-zero singular values.
+- For both: $Sigma$ is constructed s.t. $sigma_1 >= ... >= sigma_k >= 0$. Rank: number of non-zero singular values.