### 2.75 Find orthonormal vectors that span the range of matrix A

Problem: Given the matrix $$A$$ whose columns represents some vectors, ﬁnd the set of orthonormal vectors that span the same space as $$A$$ and verify the result. Let $A=\begin {bmatrix} 0 & 1 & 1 & 2\\ 1 & 2 & 3 & 4\\ 2 & 0 & 2 & 0 \end {bmatrix}$

Notice that $$A$$ has rank 2, so we should get no more than 2 vectors in the orthonormal set.

With MATLAB use the orth(A) function, With Mathematica, use {u,s,v}=SingularValueDecomposition[A] , and since the rank is 2, then the ﬁrst 2 columns of matrix u will give the answer needed (any 2 columns of u will also give a set of orthonormal vectors).

 Mathematica Remove["Global*"]; mat = {{0, 1, 1, 2}, {1, 2, 3, 4}, {2, 0, 2, 0}}; r = MatrixRank[mat]  2 {u,s,v}=SingularValueDecomposition[mat]; orth=N[u[[All,{1,r}]]]   {{0.378151, -0.308379}, {0.887675, -0.146825}, {0.262747, 0.939864}}  Chop[Transpose[orth].orth]  {{1.,0}, {0,1.}} 

 Matlab clear all; A=[0 1 1 2 1 2 3 4 2 0 2 0]; R=orth(A)  R = -0.3782 0.3084 -0.8877 0.1468 -0.2627 -0.9399  R'*R   1.0000 0.0000 0.0000 1.0000 `