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Chapter 6: Q6.2-26E (page 331)
Question: Suppose W is a subspace of \({\mathbb{R}^n}\) spanned by a nonzero orthogonal vectors. Explain why \(W = {\mathbb{R}^n}\).
By Theorem 4, the orthogonal vectors must be linearly independent, so the set spans W and it is the basis for W. Thus, \(W = {\mathbb{R}^n}\).
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Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be a linear transformation that preserves lengths; that is, \(\left\| {T\left( {\bf{x}} \right)} \right\| = \left\| {\bf{x}} \right\|\) for all x in \({\mathbb{R}^n}\).
Let \({{\bf{u}}_1},......,{{\bf{u}}_p}\) be an orthogonal basis for a subspace \(W\) of \({\mathbb{R}^n}\), and let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be defined by \(T\left( x \right) = {\rm{pro}}{{\rm{j}}_W}x\). Show that \(T\) is a linear transformation.
In Exercises 9-12, find a unit vector in the direction of the given vector.
12. \(\left( {\begin{array}{*{20}{c}}{\frac{8}{3}}\\2\end{array}} \right)\)
To measure the take-off performance of an airplane, the horizontal position of the plane was measured every second, from \(t = 0\) to \(t = 12\). The positions (in feet) were: 0, 8.8, 29.9, 62.0, 104.7, 159.1, 222.0, 294.5, 380.4, 471.1, 571.7, 686.8, 809.2.
a. Find the least-squares cubic curve \(y = {\beta _0} + {\beta _1}t + {\beta _2}{t^2} + {\beta _3}{t^3}\) for these data.
b. Use the result of part (a) to estimate the velocity of the plane when \(t = 4.5\) seconds.
In Exercises 17 and 18, all vectors and subspaces are in \({\mathbb{R}^n}\). Mark each statement True or False. Justify each answer.
a. If \(W = {\rm{span}}\left\{ {{x_1},{x_2},{x_3}} \right\}\) with \({x_1},{x_2},{x_3}\) linearly independent,
and if \(\left\{ {{v_1},{v_2},{v_3}} \right\}\) is an orthogonal set in \(W\) , then \(\left\{ {{v_1},{v_2},{v_3}} \right\}\) is a basis for \(W\) .
b. If \(x\) is not in a subspace \(W\) , then \(x - {\rm{pro}}{{\rm{j}}_W}x\) is not zero.
c. In a \(QR\) factorization, say \(A = QR\) (when \(A\) has linearly
independent columns), the columns of \(Q\) form an
orthonormal basis for the column space of \(A\).
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