Orthonormal Sets

Question

Find a vector V→4 in V4 such that the vectors V→1, V→2, V→3, and V→4 are orthonormal.

v1 = 0.5, 0.5, 0.5, 0.5

v2 = -0.5, 0.5, 0.5, -0.5

v3 = 0.5, 0.5, -0.5 ,-0.5

Answer 1

Let's make a m x n matrix A where each vector is a row i.e.

0.5 0.5 0.5 0.5

A = -0.5 0.5 0.5 -0.5

0.5 0.5 -0.5 -0.5

where m = 3 is number of rows and n = 4 is number of components i.e. dimension of vector space. We have to find vector x (=v4) such that A * x = 0 i.e. all dot (scalar) products between x and v (1..3) are zero. Multiplying both sides by transpose of A gives us:

A^T A x = 0

i.e. is an eigenvector of 4 x 4 matrix A^T A such that it's eigenvalue is 0. Solving this eigenvalue problem gives us a solution:

v4 = x = [-0.5, 0.5, -0.5, 0.5]