Tom's Sept 5 edits of two lectures

Author: thomassargent30

lectures/svd_intro.md

@@ -466,7 +466,7 @@ print(f'rank of X = {rr}')
 **Properties:**
 
 * Where $U$ is constructed via a full SVD, $U^\top  U = I_{p\times p}$ and  $U U^\top  = I_{m \times m}$
-* Where $\hat U$ is constructed via a reduced SVD, although $\hat U^\top  \hat U = I_{p\times p}$ it happens that  $\hat U \hat U^\top  \neq I_{m \times m}$
+* Where $\hat U$ is constructed via a reduced SVD, although $\hat U^\top  \hat U = I_{p\times p}$, it happens that  $\hat U \hat U^\top  \neq I_{m \times m}$
 
 We illustrate these properties for our example with the following code cells.
 
@@ -703,7 +703,7 @@ provided that  we set
 
 * ${V_k}^{T}=\tilde{\epsilon_k}$ (the $k$th principal component)
 
-Because  there are alternative algorithms for  computing  $P$ and $U$ for  given a data matrix $X$, depending on  algorithms used, we might have sign differences or different orders between eigenvectors.
+Because  there are alternative algorithms for  computing  $P$ and $U$ for  given a data matrix $X$, depending on  algorithms used, we might have sign differences or different orders of eigenvectors.
 
 We can resolve such ambiguities about  $U$ and $P$ by
 

lectures/wald_friedman.md

@@ -57,7 +57,7 @@ Key ideas in play will be:
 - Dynamic programming
 - Type I and type II statistical errors
     - a type I error occurs when you reject a null hypothesis that is true
-    - a type II error is when you accept a null hypothesis that is false
+    - a type II error occures when you accept a null hypothesis that is false
 - Abraham Wald's **sequential probability ratio test**
 - The **power** of a statistical test
 - The **critical region** of a statistical test