fix project 1 q1 bug

project1/Project1_Q1.py

@@ -38,9 +38,13 @@ def proj_SVD(A:np.ndarray, b:np.ndarray) -> np.ndarray:
     = U S V^* (V S^2 V^*)^(-1) V S U^* b
     = U S V^* V S^(-2) V^* V S U^* b
     = U U^* b
+    If A = U S V^*, then the projection onto the column space of A is:
+        proj_A(b) = U_r U_r^* b
+    where U_r are the left singular vectors corresponding to nonzero singular values.
     """
+    U_r = U[:, :np.linalg.matrix_rank(A)]  # Take only the relevant columns
     # Compute the projection using the SVD components
-    projection = U @ U.conj().T @ b
+    projection = U_r @ U_r.conj().T @ b
     return projection
 
 def build_A(eps: float) -> np.ndarray: