Orthogonal and orthonormal basis

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Let W be the subspace of V4(R) spanned by the following set of vectors:

Use the Gram-Schmidt orthogonalisation process to find an orthogonal basis for W. Find also an orthonormal basis for W.

The Gram-Schmidt Theorem:
Given a basis for subspace W of Vn define:

Then is an orthogonal basis of W and has the same span as

We shall use the standard inner product:

In our case:

And:

In the next step we have:

Continued on next page

Thus the orthogonal base is:

The orthonormal basis is defined as the basis vectors divided by their length:

We already know that:

Therefore the orthonormal basis is:

In column form:

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