Chapter 6: Q24E (page 331)
Find a formula for the least-squares solution of\(Ax = b\)when the columns of A are orthonormal.
The formula for the least-square solution is \(\hat x = {A^T}b\).
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Find an orthogonal basis for the column space of each matrix in Exercises 9-12.
11. \(\left( {\begin{aligned}{{}{}}1&2&5\\{ - 1}&1&{ - 4}\\{ - 1}&4&{ - 3}\\1&{ - 4}&7\\1&2&1\end{aligned}} \right)\)
Let \(U\) be an \(n \times n\) orthogonal matrix. Show that if \(\left\{ {{{\bf{v}}_1}, \ldots ,{{\bf{v}}_n}} \right\}\) is an orthonormal basis for \({\mathbb{R}^n}\), then so is \(\left\{ {U{{\bf{v}}_1}, \ldots ,U{{\bf{v}}_n}} \right\}\).
Given \(A = QR\) as in Theorem 12, describe how to find an orthogonal\(m \times m\)(square) matrix \({Q_1}\) and an invertible \(n \times n\) upper triangular matrix \(R\) such that
\(A = {Q_1}\left[ {\begin{aligned}{{}{}}R\\0\end{aligned}} \right]\)
The MATLAB qr command supplies this “full” QR factorization
when rank \(A = n\).
Exercises 19 and 20 involve a design matrix \(X\) with two or more columns and a least-squares solution \(\hat \beta \) of \({\bf{y}} = X\beta \). Consider the following numbers.
(i) \({\left\| {X\hat \beta } \right\|^2}\)—the sum of the squares of the “regression term.” Denote this number by .
(ii) \({\left\| {{\bf{y}} - X\hat \beta } \right\|^2}\)—the sum of the squares for error term. Denote this number by \(SS\left( E \right)\).
(iii) \({\left\| {\bf{y}} \right\|^2}\)—the “total” sum of the squares of the \(y\)-values. Denote this number by \(SS\left( T \right)\).
Every statistics text that discusses regression and the linear model \(y = X\beta + \in \) introduces these numbers, though terminology and notation vary somewhat. To simplify matters, assume that the mean of the -values is zero. In this case, \(SS\left( T \right)\) is proportional to what is called the variance of the set of -values.
19. Justify the equation \(SS\left( T \right) = SS\left( R \right) + SS\left( E \right)\). (Hint: Use a theorem, and explain why the hypotheses of the theorem are satisfied.) This equation is extremely important in statistics, both in regression theory and in the analysis of variance.
In exercises 1-6, determine which sets of vectors are orthogonal.
What do you think about this solution?
We value your feedback to improve our textbook solutions.
