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QUESTION

Apply the Gram-Schmidt process to the following vectors:

$(1,2,3,4), (4,3,2,6), (18,11,4,22).$

Explain your answer.

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ANSWER The process gives

\begin{eqnarray*}
\textbf{w}_1&=&(1,2,3,4)\\
\textbf{w}_2&=&(4,3,2,6)-\frac{\{(4,3,2,6).(1,2,3,4)\}(1,2,3,4)}{30}\\
&=&(4,3,2,6)-\frac{4}{3}(1,2,3,4)
\end{eqnarray*}

For an orthonormal basis it is simpler to use

$3\textbf{w}_2=3(4,3,2,6)-4(1,2,3,4)=(8,1,-6,2)$

so

\begin{eqnarray*}
\textbf{w}_3&=&(18,11,4,22)-\frac{\{(18,11,4,22).(8,1,-6,2)\}(8,1,-6,2)}{105}\\
&-&\frac{\{(18,11,4,22).(1,2,3,4)\}(1,2,3,4)}{30}\\
&=&(18,11,4,22)-\frac{175}{105}(8,1,-6,2)-\frac{140}{30}(1,2,3,4)\\
&=&(18,11,4,22)-\frac{5}{3}(8,1,-6,2)-\frac{14}{3}(1,2,3,4)\\
3\textbf{w}_3&=&(54,33,12,66)-(40,5,-30,10)-(14,28,42,56)\\&=&(0,0,0,0).
\end{eqnarray*}

The explanation is that the three given vectors are linearly
dependent.



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