Merge pull request #511 from mimi2626/patch-3

Update learn.md

Author: moe18

questions/7_matrix-transformation/learn.md

@@ -1,12 +1,13 @@
 ### Matrix Transformation using $T^{-1} A S$
 
-Transforming a matrix $A$ using the operation $T^{-1} A S$ involves several steps. This operation changes the basis of matrix $A$ using two matrices $T$ and $S$, with $T$  being invertible.
+Transforming a matrix $A$ using the operation $T^{-1} A S$ involves several steps. This operation changes the basis of matrix $A$ using two matrices $T$ and $S$, with $T$ and $S$ being invertible, to avoid loss of information.
+(Multiplying by non-invertible $S$ would result in a loss of dimensions)
 
 ### Steps for Transformation
 
 Given matrices $A$, $T$, and $S$:
 
-1. **Check Invertibility**: Verify that $T$ is invertible by ensuring its determinant is non-zero; otherwise, return $-1$.
+1. **Check Invertibility**: Verify that $T$ and $S$ are invertible by ensuring their determinants are non-zero; otherwise, return $-1$.
 2. **Compute Inverses**: Find the invers of $T$, denoted as $T^{-1}$.
 3. **Perform Matrix Multiplication**: Calculate the transformed matrix:
 
@@ -45,6 +46,7 @@ $$
 #### Check Invertibility:
 
 - $\det(T) = 4 \neq 0$
+- $\det(S) = 1 \neq 0$
 
 #### Compute Inverses: