Suppose 𝐮1, …, 𝐮k form an orthonormal set of vectors in ℝ^n with k<n. Let Q=(𝐮1 𝐮2 …𝐮k) denote t

step 1 In this problem, we have an orthogonal set of vectors, and we are told that these vectors are al

Suppose 𝐮1, …, 𝐮k form an orthonormal set of vectors in ℝ^n...

Suppose $\mathbf{u}_1, \ldots, \mathbf{u}_k$ form an orthonormal set of vectors in $\mathbb{R}^n$ with $k<n$. Let $Q=\left(\mathbf{u}_1 \mathbf{u}_2 \ldots \mathbf{u}_k\right)$ denote the $n \times k$ matrix whose columns are the orthonormal vectors. (a) Prove that $Q^T Q=\mathrm{I}_k$. (b) Is $Q Q^T=\mathrm{I}_n$ ?

Suppose $\mathbf{u}_1, \ldots, \mathbf{u}_k$ form an orthonormal set of vectors in $\mathbb{R}^n$ with $k<n$. Let $Q=\left(\mathbf{u}_1 \mathbf{u}_2 \ldots \mathbf{u}_k\right)$ denote the $n \times k$ matrix whose columns are the orthonormal vectors.
(a) Prove that $Q^T Q=\mathrm{I}_k$. (b) Is $Q Q^T=\mathrm{I}_n$ ?

Key Concepts

Orthonormal Vectors

An orthonormal set of vectors in a Euclidean space is a set where each vector has unit length and any two distinct vectors are orthogonal. This property guarantees that the inner product of any two distinct vectors in the set is zero and the inner product of a vector with itself is one, which is fundamental in simplifying many matrix operations in linear algebra.

Matrix Transpose and Multiplication

The transpose of a matrix is obtained by switching its rows and columns. When multiplying a matrix by its transpose, the resultant entries are dot products of the original matrix’s columns. In the case of a matrix with orthonormal columns, this multiplication simplifies such that the dot product yields an identity matrix, hence showing properties like Q^T Q = I.

Identity Matrix

The identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix multiplication. In this context, showing that Q^T Q equals the identity matrix emphasizes that the columns of Q are not only linearly independent but also orthonormal.

Projection Matrix

A projection matrix maps vectors onto a subspace. When a non-square matrix with orthonormal columns, like Q, is multiplied as Q Q^T, the resulting matrix represents the projection onto the column space of Q. This projection matrix equals the identity only if the subspace is the entire space, which is not the case when the number of orthonormal vectors k is less than the dimension n.

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