Create the program to return the orthogonal basis and orthonormal basis

Write a program to input any number of vectors in R^n and return return the orthogonal basis and orthonormal basis of the subspace spanned by these vectors using Gram Schmidt process. Can someone check if I do right

for j= 1:n

v = A (: , j) ;

for i = 1: j-1

R(i,j) = Q (:, i )' * A ( :, j) ;

v = v - R ( i ,j ) * Q ( :, i);

end

R ( j, j ) = norm ( v );

Q (:, j) = v/R ( j, j ) ;

end

Answer 1

[Torsten]

As far as I can see, you only return the orthonormal basis from the Gram Schmidt process in the matrix Q.

And saving the norms is superfluous if you have to return the orthonormal basis, too.

n = 2;
A = [1 3; 4 -7; -1 -12];
Q = zeros(size(A));
QN = zeros(size(A));
for j= 1:n 
    v = A (:,j) ; 
    for i = 1: j-1
        rij = Q(:,i).' * A(:,j) / (Q(:,i).' * Q(:,i));
        v = v - rij * Q(:,i);
    end
    Q(:,j) = v;
    QN(:,j) = v/norm(v);
end
sqrt(Q.'*Q)

ans = 2×2

4.2426 0 0 13.8784

QN.'*QN

ans = 2×2

1.0000 0 0 1.0000

[Nguyet Nguyen]

tks u so much

[Torsten]

If the resulting vector v is the zero vector, you shouldn't include it as column in Q. This will happen if vi is already in the span of v1,...,v_i-1. And this will definitely happen if more than n vectors are supplied - as is possible according to the formulation of the assignment.

Answer 2

[Matt J]

You should just use orth to get the orthonormal basis. And you should use qr instead of Gram-Schmidt.