Expressing a vector as a linear combination of an orthonormal set of vectors

Question

In solution to second question, i'm struggling to find the correct coefficients with my method of creating an augmented matrix... could somebody maybe hint or show how they are getting these coefficients?

Answer 1

There is a VERY useful theorem regarding orthonormal bases and that is:

Let $\mathbf{u}_1, \ldots, \mathbf{u}_n$ be an orthonormal basis of some space $V$ and let $\mathbf{v} = a_1 \mathbf{u}_1 + \ldots a_n \mathbf{u}_n$ be the decomposition of some vector $\mathbf{v} \in V$ according to this basis. Then the coefficient $a_i$ equals the inner product of $\mathbf{u}_i$ with $\mathbf{v}$.

The inner product is also sometimes called scalar product or (in the setting of $\mathbb{R}^n$) dot product. It is the thing that gives meaning to the notions of 'orthogonal' and 'orthonormal'.

The interesting question is of course WHY this theorem is true. The answer to this depends on how the inner product was defined/introduced. Look that up in your text and then you will probably be able to prove the above theorem yourself.