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Chapter 6: Q31SE (page 331)

Show that if \(x\) is in both \(W\) and \({W^ \bot }\), then \(x = 0\).

Short Answer

Expert verified

It is proved that \(x = 0\).

Step by step solution

01

Definition of Orthogonality

Two vectors \(u\) and \(v\) in \({\mathbb{R}^n}\) are orthogonal to each other if \(u \cdot v = 0\).

02

Proof

A vector \(x\) is in \({W^ \bot }\) if and only if \(x\) is orthogonal to every vector in a set that spans \(W\).

Since, \(x\) in in \(W\) also. Therefore, it must be orthogonal to itself.

That means, \(x \cdot x = 0\). This can be true if \(x = 0\).

Hence, it is proved that \(x = 0\).

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Most popular questions from this chapter

In Exercises 1-6, the given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.

3. \(\left( {\begin{aligned}{{}{}}2\\{ - 5}\\1\end{aligned}} \right),\left( {\begin{aligned}{{}{}}4\\{ - 1}\\2\end{aligned}} \right)\)

Let \(X\) be the design matrix in Example 2 corresponding to a least-square fit of parabola to data \(\left( {{x_1},{y_1}} \right), \ldots ,\left( {{x_n},{y_n}} \right)\). Suppose \({x_1}\), \({x_2}\) and \({x_3}\) are distinct. Explain why there is only one parabola that best, in a least-square sense. (See Exercise 5.)

Find an orthonormal basis of the subspace spanned by the vectors in Exercise 3.

Find the distance between \({\mathop{\rm x}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{10}\\{ - 3}\end{aligned}} \right)\) and \({\mathop{\rm y}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{ - 1}\\{ - 5}\end{aligned}} \right)\).

Compute the quantities in Exercises 1-8 using the vectors

\({\mathop{\rm u}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{ - 1}\\2\end{aligned}} \right),{\rm{ }}{\mathop{\rm v}\nolimits} = \left( {\begin{aligned}{*{20}{c}}4\\6\end{aligned}} \right),{\rm{ }}{\mathop{\rm w}\nolimits} = \left( {\begin{aligned}{*{20}{c}}3\\{ - 1}\\{ - 5}\end{aligned}} \right),{\rm{ }}{\mathop{\rm x}\nolimits} = \left( {\begin{aligned}{*{20}{c}}6\\{ - 2}\\3\end{aligned}} \right)\)

8. \(\left\| {\mathop{\rm x}\nolimits} \right\|\)
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