An orthogonal basis for a vector space is a set of vectors that are both:

1. Orthogonal: Each pair of vectors in the set is perpendicular (orthogonal) to each other. Mathematically, if $\{ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \}$ is the set of vectors, then for any two different vectors $\mathbf{v}_i$ and $\mathbf{v}_j$ in the set, their dot product is zero:
   $\mathbf{v}_i \cdot \mathbf{v}_j = 0 \quad \text{for } i \neq j.$

2. Spanning: The set of vectors spans the vector space, meaning any vector in the space can be represented as a linear combination of the vectors in this set.

An orthonormal basis is a special case of an orthogonal basis in which each vector is also of unit length, i.e., $\|\mathbf{v}_i\| = 1$. This adds the property that
$\mathbf{v}_i \cdot \mathbf{v}_i = 1.$

Finding an Orthogonal Basis

If given a basis that isn’t orthogonal, we can apply the Gram-Schmidt process to transform it into an orthogonal (or orthonormal) basis.

Would you like further details on the Gram-Schmidt process or have other questions on this topic?

Related Questions

1. What is the Gram-Schmidt process, and how does it create an orthogonal basis?
2. How do you determine if a set of vectors is orthogonal?
3. Can a non-square matrix have an orthogonal basis?
4. How is an orthonormal basis used in practical applications, like in Fourier series?
5. Why is orthogonality important in vector spaces and linear transformations?

Tip

When working with an orthogonal basis, computations like projections become simpler due to the perpendicularity of the vectors involved.