This page outlines a chapter on solving matrix equations \(Ax=b\), emphasizing orthogonality for approximate solutions. It begins with definitions in Sections 6.1 and 6.2, discusses orthogonal project...This page outlines a chapter on solving matrix equations \(Ax=b\), emphasizing orthogonality for approximate solutions. It begins with definitions in Sections 6.1 and 6.2, discusses orthogonal projections for finding closest vectors in Section 6.3, and introduces the least-squares method in Section 6.5, highlighting its applications in data modeling, including predicting best-fit lines or ellipses in historical astronomical data.
In other words, \(A\hat x\) is the vector whose entries are the \(y\)-coordinates of the graph of the line at the values of \(x\) we specified in our data points, and \(\vec{b}\) is the vector whose e...In other words, \(A\hat x\) is the vector whose entries are the \(y\)-coordinates of the graph of the line at the values of \(x\) we specified in our data points, and \(\vec{b}\) is the vector whose entries are the \(y\)-coordinates of those data points.
