Suppose 𝐯1, …, 𝐯k are nonzero mutually orthogonal elements of an inner product space V. Write do

step 1 Okay, this question gives us that U and V are orthogonal matrices, and it wants us to show that

Suppose 𝐯1, …, 𝐯k are nonzero mutually orthogonal elements...

Key Concepts

Inner Product Space

An inner product space is a vector space equipped with an inner product, a function that assigns a scalar to any pair of vectors and satisfies properties like linearity, symmetry, and positive-definiteness. This structure allows the definition of concepts such as length and angle, which are fundamental in understanding geometric relations in the space.

Gram Matrix

The Gram matrix is constructed from a set of vectors in an inner product space by taking the inner products of each pair of vectors as its entries. It encapsulates the geometric relationships among the vectors, including lengths and mutual orientations, and plays an important role in determining properties like linear independence and orthogonality.

Orthogonality

Orthogonality in an inner product space denotes that the inner product of any two distinct vectors is zero. Mutually orthogonal vectors not only simplify the computation of the Gram matrix by yielding zeros off the diagonal, but also ensure that the vectors are linearly independent, which is crucial for many applications in linear algebra and geometry.

Nonsingularity

A matrix is nonsingular if it is invertible, meaning it has full rank and a nonzero determinant. When the set of vectors is mutually orthogonal and nonzero, the resulting Gram matrix is diagonal with nonzero entries along the diagonal, guaranteeing that its determinant is the product of these nonzero entries. This confirms that the Gram matrix is invertible (nonsingular), which is essential for many computations in linear algebra.

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