Linear Algebra?

Question

let vi=ui / ||ui|| for i=1,2,3 u is a members of N3, Show that
u=v1 + v2 + v3. How to proof it.

Answer 1

I'm assuming ui ≠ 0 for i = 1, 2, 3, to make sense of the definition. We have

u = u1 + u2 + u3 = ||u1|| v1 + ||u2|| v2 + ||u3|| v3.

Now, = = / ||ui||

= [ + + ] / ||ui||

= / ||ui|| (since the basis is orthogonal)

= ||ui||² / ||ui|| (by the definition of the norm)

= ||ui||,

for i = 1, 2, 3.

Hence, ||ui|| = , and substituting this into our first equation yields

u = v1 + v2 + v3,

which is what we wished to prove.