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Section 7.6 Gram-Schmidt and QR-Factorization: Homework
Exercises Exercises
1.
Exercise Group.
For each set of vectors, \(S\text{,}\)
Orthogonalize \(S\) by the Gram-Schmidt process (i.e, find the \(\vec{u_i}\)’s).
Form \(A\) using the vectors in \(S\) as columns, and write the \(QR\)-factorization of \(A\text{.}\)
2.
\(S=\left\{\left(\begin{array}{r}-8\\8\\4 \end{array} \right), \left(\begin{array}{rrr}20\\-32\\-4 \end{array} \right)\left(\begin{array}{rrr}4\\20\\4 \end{array} \right)\right\}\text{.}\)
3.
\(S=\left\{\left(\begin{array}{r}-15\\-6\\-10 \end{array} \right), \left(\begin{array}{rrr}20\\27\\26 \end{array} \right)\left(\begin{array}{rrr}41\\-71\\-55 \end{array} \right)\right\}\text{.}\)
4.
\(S=\left\{\left(\begin{array}{r}2\\8\\16 \end{array} \right), \left(\begin{array}{r}20\\8\\34 \end{array} \right)\left(\begin{array}{r}52\\-26\\-34 \end{array} \right)\right\}\text{.}\)
5.
\(S=\left\{\left(\begin{array}{r}-29\\26\\2 \end{array} \right), \left(\begin{array}{r}-44\\65\\38 \end{array} \right)\left(\begin{array}{r}79\\26\\47 \end{array} \right)\right\}\text{.}\)
