Let A be an m ×n matrix with ker A={0}. Suppose that we use the Gram-Schmidt algorithm to factor

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Let A be an m ×n matrix with ker A={0}. Suppose that we use...

Let $A$ be an $m \times n$ matrix with $\operatorname{ker} A=\{0\}$. Suppose that we use the Gram-Schmidt algorithm to factor $A=Q R$ as in Exercise 4.3.32. Prove that the least squares solution to the linear system $A \mathbf{x}=\mathbf{b}$ is found by solving the triangular system $R \mathbf{x}=Q^T \mathbf{b}$ by Back Substitution.

Key Concepts

Gram-Schmidt Process

The Gram-Schmidt process is a method for converting a set of linearly independent vectors into an orthonormal set of vectors. This technique is crucial when one needs to form an orthonormal basis for the column space of a matrix, which is essential in various computational methods, including QR factorization.

QR Factorization

QR factorization decomposes a matrix into the product of an orthogonal matrix Q and an upper triangular matrix R. This factorization simplifies many problems in numerical linear algebra, such as solving linear systems and determining the least squares solution when the system is over-determined.

Least Squares Approximation

The least squares approximation is the approach of finding an approximate solution to an over-determined system by minimizing the sum of the squares of the residuals. This method yields the solution that best fits the data in the sense of minimizing the Euclidean distance between the observed vector and its projection onto the column space of the matrix.

Back Substitution

Back substitution is a systematic procedure for solving a triangular system of linear equations. In the context of QR factorization, once the system is reduced to solving R x = Q^T b, the upper triangular nature of R allows one to efficiently compute the solution by solving for the variables starting from the last equation and proceeding upwards.

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