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Worksheet 7.5.1 Worksheet
A4 US

1.

Prove Theorem 7.4.1.

2.

Prove Corollary 7.4.3.

3.

Prove Theorem 7.4.12.

In this worksheet we will derive the process by which one may create an orthogonal basis \(\left\{\vec{u_1},\vec{u_2},\vec{u_3}\right\}\) of \(\R^3\) from a linearly independent basis \(\left\{\vec{v_1},\vec{v_2},\vec{v_3}\right\}\) of \(\R^3\text{.}\)

Consider the set \(S:=\left\{\vec{v_1},\vec{v_2},\vec{v_3}\right\}=\left\{\left(\begin{array}{r}2\\8\\16 \end{array} \right), \left(\begin{array}{rrr}20\\8\\34 \end{array} \right)\left(\begin{array}{rrr}52\\-26\\-34 \end{array} \right)\right\}\text{.}\) Show that \(S\) is linearly independent in any way you like.
Write \(\vec{u_1}=\vec{v_1}\text{.}\) Then subtract from \(\vec{v_2}\) its projection to span\(\{\vec{v_1}\}\text{,}\) call the result \(\vec{u_2}\text{,}\) and denote \(\displaystyle{\widehat{x}_{12}=\frac{\vec{u_1}^T\vec{v_2}}{\vec{u_1}^T\vec{u_1}}}\text{.}\) At this point you have two of the three vectors you need.
Next subtract from \(\vec{v_3}\) its projections to span\(\{\vec{u_1}\}\) and to span\(\{\vec{u_2}\}\text{,}\) and call the result \(\vec{u_3}\text{,}\) writing

\begin{equation*} \widehat{x}_{13}=\frac{\vec{u_1}^T\vec{v_3}}{\vec{u_1}^T\vec{u_1}}, \widehat{x}_{23}=\frac{\vec{u_2}^T\vec{v_3}}{\vec{u_2}^T\vec{u_2}}\text{.} \end{equation*}
Verify that \(\left\{\vec{u_1},\vec{u_2},\vec{u_3}\right\}\) is orthogonal.
Construct \(R=\left(\begin{array}{ccc}1\amp \widehat{x}_{12}\amp \widehat{x}_{13}\\0\amp 1\amp \widehat{x}_{23}\\0\amp 0\amp 1 \end{array} \right)\) and \(Q=\begin{pmatrix}|\amp |\amp \cdots\amp |\\\vec{u_1}\amp \vec{u_2}\amp \cdots\amp \vec{u_n}\\|\amp |\amp \cdots\amp | \end{pmatrix}\) and verify that \(A=\left(\begin{array}{rrr}2\amp 8\amp 16\\20\amp 8\amp 34\\52\amp -26\amp -34 \end{array} \right)=QR\text{.}\)
Repeat your work for \(T:=\left\{\vec{v_1},\vec{v_2},\vec{v_3}\right\}=\left\{\left(\begin{array}{r}-8\\-8\\4 \end{array} \right), \left(\begin{array}{rrr}-12\\0\\12 \end{array} \right)\left(\begin{array}{rrr}20\\20\\-28 \end{array} \right)\right\}\text{.}\)

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