--- a/pca.m
+++ b/pca.m
@@ -1,31 +1,9 @@
 function [U, S] = pca(X)
-%PCA Run principal component analysis on the dataset X
-%   [U, S, X] = pca(X) computes eigenvectors of the covariance matrix of X
-%   Returns the eigenvectors U, the eigenvalues (on diagonal) in S
-%
-
-% Useful values
-[m, n] = size(X);
-
-% You need to return the following variables correctly.
-U = zeros(n);
-S = zeros(n);
+  ##PCA Run principal component analysis on the dataset X
+  ##   [U, S, X] = pca(X) computes eigenvectors of the covariance matrix of X
+  ##   Returns the eigenvectors U, the eigenvalues (on diagonal) in S
+  ##
 
-% ====================== YOUR CODE HERE ======================
-% Instructions: You should first compute the covariance matrix. Then, you
-%               should use the "svd" function to compute the eigenvectors
-%               and eigenvalues of the covariance matrix. 
-%
-% Note: When computing the covariance matrix, remember to divide by m (the
-%       number of examples).
-%
+  [U, S, ~] = svd (X'*X/rows (X));
 
-
-
-
-
-
-
-% =========================================================================
-
-end
+endfunction

--- a/projectData.m
+++ b/projectData.m
@@ -1,26 +1,11 @@
 function Z = projectData(X, U, K)
-%PROJECTDATA Computes the reduced data representation when projecting only 
-%on to the top k eigenvectors
-%   Z = projectData(X, U, K) computes the projection of 
-%   the normalized inputs X into the reduced dimensional space spanned by
-%   the first K columns of U. It returns the projected examples in Z.
-%
-
-% You need to return the following variables correctly.
-Z = zeros(size(X, 1), K);
+  ##PROJECTDATA Computes the reduced data representation when projecting only 
+  ##on to the top k eigenvectors
+  ##   Z = projectData(X, U, K) computes the projection of 
+  ##   the normalized inputs X into the reduced dimensional space spanned by
+  ##   the first K columns of U. It returns the projected examples in Z.
+  ##
 
-% ====================== YOUR CODE HERE ======================
-% Instructions: Compute the projection of the data using only the top K 
-%               eigenvectors in U (first K columns). 
-%               For the i-th example X(i,:), the projection on to the k-th 
-%               eigenvector is given as follows:
-%                    x = X(i, :)';
-%                    projection_k = x' * U(:, k);
-%
-
-
-
-
-% =============================================================
+  Z = X*U(:,1:K);
 
 end

--- a/recoverData.m
+++ b/recoverData.m
@@ -1,28 +1,11 @@
 function X_rec = recoverData(Z, U, K)
-%RECOVERDATA Recovers an approximation of the original data when using the 
-%projected data
-%   X_rec = RECOVERDATA(Z, U, K) recovers an approximation the 
-%   original data that has been reduced to K dimensions. It returns the
-%   approximate reconstruction in X_rec.
-%
-
-% You need to return the following variables correctly.
-X_rec = zeros(size(Z, 1), size(U, 1));
-
-% ====================== YOUR CODE HERE ======================
-% Instructions: Compute the approximation of the data by projecting back
-%               onto the original space using the top K eigenvectors in U.
-%
-%               For the i-th example Z(i,:), the (approximate)
-%               recovered data for dimension j is given as follows:
-%                    v = Z(i, :)';
-%                    recovered_j = v' * U(j, 1:K)';
-%
-%               Notice that U(j, 1:K) is a row vector.
-%               
-
-
-
-% =============================================================
+  ##RECOVERDATA Recovers an approximation of the original data when using the 
+  ##projected data
+  ##   X_rec = RECOVERDATA(Z, U, K) recovers an approximation the 
+  ##   original data that has been reduced to K dimensions. It returns the
+  ##   approximate reconstruction in X_rec.
+  ##
+  
+  X_rec = Z*U(:, 1:K)';
 
 end