Gram–Schmidt orthogonalization. An orthogonal basis for a space spanned by some vectors can be obtained using the Gram–Schmidt orthogonalization procedure. (a) Consider two linearly independent...

Gram–Schmidt orthogonalization. An orthogonal basis for a space spanned by some vectors can be obtained using the Gram–Schmidt orthogonalization procedure.

(a) Consider two linearly independent vectors v₁
and v₂. Define z₁
= v₁
and z₂
= v₂
− v₁c₂.1, where c₂.1 = (v₁v₂)/(v₁v₁). Show that z₁
and z₂
are orthogonal. Also, show that z₁
and z₂
span the same space as v₁
and v₂.

(b) Consider three linearly independent vectors v₁, v₂, and v₃. De- fine z₁
and z₂
as in

(a) and z₃
= v₃
− c₃.1z₁
− c₃.2z₂, where c₃.i = (z_iv₃)/(z_iz_i), i = 1, 2. Show that z₁, z₂, and z₃
are mutually orthogonal and span the same space as v₁, v₂, and v₃.