Picking two pivot columns

Question

I wanted to know that if i pick two pivot columns of a Matrix who has 3 pivot columns, then can any of the two columns i picked be used as an orthogonal basis?? Or should i pick all three columns in order to create a orthogonal basis?

First column consist of (1,1,-1,-1) Second column consist of (3,-1,1,-3) Third column consist of (2,6,0,4)

Answer 1

Pivot columns in the RREF are not necessarly orthogonal.

We can choose them to form a basis and then orthogonalize by Gram-Schmidt process.

Note that with 2 pivot columns we can construct a basis for a 2 dimensional subspace, thus that choiche is necessarly different from a basis with 3 pivot columns.