Define the term "orthonormal basis."

An orthonormal basis is defined as a set of vectors that are both orthogonal and unit vectors, meaning each vector has a length (or magnitude) of 1 and every pair of distinct vectors in the set is orthogonal to each other. Orthogonality ensures that the dot product of any two different vectors in the set is zero, signifying they are at right angles to each other in the vector space. The condition of being unit vectors confirms that each vector's length equals one, simplifying numerous mathematical operations, particularly in applications like projections and transformations.

In any vector space, having an orthonormal basis is beneficial because it provides a convenient framework to represent other vectors through linear combinations. Since every vector can be expressed uniquely as a combination of the basis vectors, calculations become straightforward, and the inner product retains its intuitive interpretation. This property is particularly useful in various areas of applied mathematics, physics, and engineering.

The other options do not correctly describe an orthonormal basis. For instance, a basis consisting of linearly dependent vectors would not be able to span the space uniquely, while vectors of varying lengths would not satisfy the requirement to be unit vectors. Including a zero vector contradicts the definition of a basis since a basis requires that all