Determine an orthogonal basis for the Column space for A?

Suppose each column is a vector. The column space is the span of the column vectors. The basis of the column space is the set of linearly independent vectors that span the column space.

Step 1: Reduce to reduced row echelon form: http://www.math.odu.edu/~bogacki/cgi-bin...

The reduced row echelon form is:

[1 -2 0 0 2]
[0 0 1 0 1]
[0 0 0 1 2]
[0 0 0 0 0]

For simplicity, we take the columns containing leading entries as the linearly independent columns, which are columns 1, 3 and 4. So we take columns 1, 3 and 4 in the ORIGINAL matrix A, as the basis for the column space.

Therefore, the basis for the column space is: { (1,0,-1,0) , (-1,-1,0,1) , (0,1,2,2) }

The next step is to orthogonalise it as follows: (This method is based on the Gram-Schmidt process)

Given the basis {b1, b2, b3}

Let u1 = b1

Let u2 = b2 - (b2•u1)u1

Let u3 = b3 - (b3•u1)u1 - (b3•u2)u2)

And {u1, u2, u3} is an orthogonal basis