If v1, …, vn is an orthonormal basis for C^n, the matrix...

If $v_{1}, \ldots, v_{n}$ is an orthonormal basis for $C^{n}$, the matrix with those columns is a matrix. Show that any vector $z$ equals $\left(v_{1}^{\mathrm{H}} z\right) v_{1}+\cdots+\left(v_{n}^{\mathrm{H}} z\right) v_{n}$,

Key Concepts

Orthonormal Basis

An orthonormal basis is a set of vectors that are both orthogonal to each other and of unit length. This property ensures that the dot product (or inner product in complex spaces) between any two different basis vectors is zero, and the inner product of a basis vector with itself is one. This framework greatly simplifies computations in vector spaces, as it allows any vector to be expressed uniquely in terms of these basis vectors, with coefficients that are easy to compute.

Inner Product and Coefficient Extraction

The inner product in complex vector spaces is a tool that not only measures angles and lengths but also facilitates the extraction of coefficients when decomposing a vector. When a vector is projected onto an orthonormal basis, each coefficient in the expansion is computed as the inner product between the vector and a basis vector. This direct computation is at the heart of many algorithms in linear algebra and signal processing.

Projection onto Basis Vectors

Projection in the context of vector spaces refers to the component of a vector along a particular direction. In an orthonormal basis, each term of the expansion represents the projection of the original vector onto one of the basis vectors. These projections capture the contribution of each basis direction to the original vector, and their sum reconstructs the vector completely.

Completeness of the Basis

Completeness means that the set of basis vectors spans the entire vector space, ensuring that every vector in the space can be fully represented as a linear combination of these basis vectors. The relationship between the original vector and the projections onto the basis vectors is a reflection of the completeness property, which is fundamental in understanding how or why every vector in the space can be reconstructed exactly from its components along the basis directions.

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