Help on Inner product spaces

I am reading the text book and this was shown and it says left to the reader ..... it would be more then great if some one can show me why this is true

The statement is exactly as follows

Attachment 25988

i will assume you mean a REAL inner product space (it is not true for a COMPLEX inner product space, there's a different formula).

the norm induced by an inner product <_,_> is given by:

||w|| = √(<w,w>)

if a basis {v₁,v₂} is orthonormal with respect to a given inner product, then:

<v₁,v₁> = <v₂,v₂> = 1

<v₁,v₂> = <v₂,v₁> = 0.

hence, if w = av₁+bv₂:

||w|| = √(<w,w>) = √(< av₁+bv₂, av₁+bv₂>)

= √[<av₁,av₁+bv₁> + <bv₂,av₁+bv₂>]

= √[<av₁,av₁> + <av₁,bv₂> + <bv₂,av₁> + <bv₂,bv₂>]

= √[a<v₁,av₁> + a<v₁,bv₂> + b<v₂,av₁> + b<v₂,bv₂>]

= √[a²<v₁,v₁> + ab<v₁,v₂> + ab<v₂,v₁> + b²<v₂,v₂>]

= √[a²(1) + ab(0) + ab(0) + b²(1)]

= √(a² + b²).

If saying thank you a 100 times would show you how much i appreciate this I would do it.
really thank you very much this is excellent