Gram–Schmidt Process - Example

Example

Consider the following set of vectors in R2 (with the conventional inner product)

Now, perform Gram–Schmidt, to obtain an orthogonal set of vectors:

We check that the vectors u1 and u2 are indeed orthogonal:

noting that if the dot product of two vectors is 0 then they are orthogonal.

We can then normalize the vectors by dividing out their sizes as shown above:

\mathbf{e}_2 = {1 \over \sqrt{40 \over 25}} \begin{pmatrix}-2/5\\6/5\end{pmatrix} = {1\over\sqrt{10}} \begin{pmatrix}-1\\3\end{pmatrix}.

Read more about this topic:  Gram–Schmidt Process

Famous quotes containing the word example:

    “Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
    —Marcel Proust (1871–1922)