Orthogonal Set of Unit Vectors

Let B be a set of vectors $v_1, v_2, \ldots, v_k$ such that all vectors in B have length 1 $(\|\|v_i\|\| = 1)$ for all $i$ and are orthogonal to each other $(v_i \cdot v_j = 0)$ for $i \ne j$. Show that B is an orthonormal set and prove that B is also linearly independent.

In this problem, we are exploring the concepts of orthogonal and orthonormal sets within vector spaces, which are key in understanding many aspects of linear algebra and its applications. An orthonormal set is a set of vectors that are both orthogonal to each other and have unit length. What makes orthonormal sets particularly useful is their applicability in areas such as simplifying matrix operations and forming bases for vector spaces, which can substantially ease the computation processes in various domains, including machine learning and quantum mechanics.

To solve the given problem, recognize that you are working with an orthogonal set of unit vectors. Each vector in the set has a length of one, and they are orthogonal, meaning their dot product is zero unless the vectors are identical. These properties inherently satisfy the requirements for a set to be orthonormal. Once a set is established as orthonormal, it immediately implies linear independence. Linear independence, in this context, indicates that no vector in the set can be expressed as a linear combination of the others. This property is critical in defining bases for vector spaces because it ensures that the set spans the space without any redundancy.

Understanding the proof for linear independence in orthonormal sets also assists in grasping the broader implications of these concepts in higher-dimensional spaces. By studying how orthonormal sets relate to matrix transformations, you can delve into topics such as orthogonal matrices, which preserve vector lengths and angles and are particularly useful in maintaining numerical stability in computations.