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[M] For a matrix program, the Gram-Schmidt process works better with orthonormal vectors. Starting with $\mathbf{x}_{1}, \ldots, \mathbf{x}_{p}$ as in Theorem $11,$ let $A=\left[\mathbf{x}_{1} \quad \cdots \quad x_{p}\right] .$ Suppose $Q$ is an $n \times k$ matrix whose columns form an orthonormal basis for the subspace $W_{k}$ spanned by the first $k$ columns of $A .$ Then for $\mathbf{x}$ in $\mathbb{R}^{n}, Q Q^{T} \mathbf{x}$ is the orthogonal projection of $\mathbf{x}$ onto $W_{k}$ (Theorem 10 in Section 6.3$) .$ If $\mathbf{x}_{k+1}$ is the next column of $A$ then equation $(2)$ in the proof of Theorem 11 becomes $$\mathbf{v}_{k+1}=\mathbf{x}_{k+1}-Q\left(Q^{T} \mathbf{x}_{k+1}\right)$$

[M] In MATLAB, when $A$ has $n$ columns, suitable commands are $Q = A ( : , 1 ) / \operatorname { norm } ( A ( : , 1 ) )$ The first column of $Q$ for $j = 2 : n$ $v = A ( : , j ) - Q ^ { \star } \left( Q ^ { \prime } \star A ( : , j ) \right)$ $Q ( : , j ) = v / \operatorname { norm } ( v )$ $\frac { q } { 6 }$ Add a new column to $Q$ end

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 4

The Gramâ€“Schmidt Process

Vectors

Johns Hopkins University

Missouri State University

Harvey Mudd College

University of Nottingham

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in Problem 26. We want to construct a metrics program, for example, using the club to get the victory ization for the metrics. A. We're a has number off P columns starts from the victor Exxon the Victor Xdb Using the concept of that the K plus one equals X k plus one minus que multiplied by cute principles. Multiply it by X k plus one where K is a number smaller than p. Okay, smaller than be, and the Q has a dimension off and and rose, or to blow it by key. To do so. Okay is smaller than be smaller than or equal speed smaller than be not equals. Because we have Year K plus one, we can go beyond be Let's construct our program. Start here. The first system is to define the first column in Q que has Onley Ortho. Normal columns. Then we have que equals the first column of the Matrix, a a on one divided by the north off to make it or the normal the norm, we can use the common norm off mhm off this victory. This is the first line. A second line is to make a loop for each Victor V where we we can use the loop. Full loop four okay, equals one equals one to B minus one. We said that be Hey, Will has the greatest value off the minus one. To calculate the V. The is just a victor which equals X. Welcome from the metrics e, we have the victor K plus one toe take, for example, the second victory on the third effect and so on Minus que multiplied boy Que transposed multiplied boy the victor. Mhm Oh K plus one This victim. Then we add this victor to the metric secu. We have Q equals que and we add the Ortho normal victor which is v divided by the norm which is v divided by the norm off the nor of the we add in this step we add than or so normal Victor Thank you. Then we end the full loop after the end. We have que This is the first part of the solution. But for queue, our factory ization we should calculate or oh are equals Que transpose multiplied boy, the metrics eight And this is the final answer off our problem. This is the program. If you construct this program into the lab, we'll get the answer

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