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Chapter 5: Q4E (page 245)
Let Abe anmatrix. Is the formulanecessarily true? Explain.
Yes.
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Consider the subspace of . Where . Find a basis of , and draw a sketch illustrating the formulain this case.
Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.
15.
To make a trend analysis of six evenly spaced data points, one can use orthogonal polynomials with respect to evaluation at the points \(t = - 5, - 3, - 1,\,\,1,\,\,3,{\rm{ and }}5\).
\({p_0}\left( t \right) = 1,\,\,\,\,\,\,{p_1}\left( t \right) = t,{\rm{ and }}{p_2}\left( t \right) = \frac{3}{8}{t^2} - \frac{{35}}{8}\)
(The polynomial \({p_2}\) has been scaled so that its values at the evaluation points are small integers.)
Find the least-squares line \(y = {\beta _0} + {\beta _1}x\) that best fits the data \(\left( { - 2,0} \right),\left( { - 1,0} \right),\left( {0,2} \right),\left( {1,4} \right),{\rm{ and }}\left( {2,4} \right)\), assuming that the first and last data points are less reliable. Weight them half as much as the three interior points.
Consider a consistent system .
(a) Show that this system has a solution in .
(b) Show that the system has only one solution in .
(c) If is the solution in androle="math" localid="1660124695419" is another solution of the system , show that . The vector is called the minimal solution of the linear system .
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