Linear Algebra Find an orthogonal basis for the column space

Linear Algebra

Find an orthogonal basis for the column space of the matrix A=matrix

Solution

given basis is v1 = [6 2 -2 6]

v2 = [1 1 -2 8];

v3 = [-5 1 5 -7];

to make orthogonal basis,we use gram-schmid process

u1 = v1

u2 = v2 - <v2,u1> / (<u1,u1>).*u1

= [1 1 -2 8] - 60/80 * [6 2 -2 6] = [-7/2 -1/2 -1/2 7/2]

u3 = v3 - <v3,u1> /<u1,u1> u1 - <v3,u2> /<u2,u2> u2

= [-5 1 5 -7 ] - (-80/80)* [ 6 2 -2 6] - (-10) /25 [-7/2 -1/2 -1/2 7/2]

= [-5 1 5 -7 ]   + [ 6 2 -2 6] + 2/5* [-7/2 -1/2 -1/2 7/2]

= [12/5 14/5 -14/5 12/5]

=