Question

Consider the subspace U of R⁴ spanned by the vectors.

V₁ =(1,1,1,1), V₂= (1,1,2,4 ), V₃= (1,2,-4,-3)

Find

1) an orthogonal bassis of V.

2) an orthogonal basis of V.

Answer 1

1) An orthogonal basis:

$x_1=v_1=(1,1,1,1)$

$x_2=v_2-\frac{v_2\cdot x_1}{x_1\cdot x_1}x_1=(1,1,2,4 )-\frac{8}{4}(1,1,1,1)=(-1,-1,0,2)$

$x_3=v_3-\frac{v_3\cdot x_1}{x_1\cdot x_1}x_1-\frac{v_3\cdot x_2}{x_2\cdot x_2}x_2$

$x_3= (1,2,-4,-3)+\frac{4}{4}(1,1,1,1)-\frac{-1-2-6}{6}(-1,-1,0,2)=(0.5,1.5,-3,1)$

2) An orthonormal basis:

$w_1=\frac{x_1}{||x_1||}=\frac{1}{2}(1,1,1,1)$

$w_2=\frac{x_2}{||x_2||}=\frac{1}{\sqrt{6}}(-1,-1,0,2)$

$w_3=\frac{x_3}{||x_3||}=\frac{1}{\sqrt{12.5}}(0.5,1.5,-3,1)=\frac{\sqrt{2}}{5}(0.5,1.5,-3,1)$