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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q27E (page 247)

Consider an isomorphism linear system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ , where A is $3 脳 2$ matrix. The least square solution of this system is $\overset{鈫�}{x} = [\begin{matrix} 7 \\ 11 \end{matrix}]$ . Consider an orthogonal $3 脳 3$ matrix S. Find the least square solutions of the system $SA \overset{鈫�}{x} = S \overset{鈫�}{b}$ .

Short Answer

Expert verified

The solution is $\overset{鈫�}{x} = [\begin{matrix} 7 \\ 11 \end{matrix}]$ .

Step by step solution

01

Step:1 Explanation of the solution

Consider the inconsistent linear system $A = \overset{鈫�}{x} = \overset{鈫�}{b}$ , the least square solution is as follows.

$\overset{鈫�}{x} = [\begin{matrix} 7 \\ 11 \end{matrix}]$ , where A is the $3 脳 2$ matrix.

Since, S is the orthogonal $3 脳 3$ matrix.

Consider S is the orthogonal $3 脳 3$ matrix.

If $\overset{鈫�}{x}$ is the least square solution of the system $A = \overset{鈫�}{x} = \overset{鈫�}{b}$ .

Then if $\overset{鈫�}{x}$ is also least square solution of the system $SA \overset{鈫�}{x} = S \overset{鈫�}{b}$ .

Hence, the least square solution of the system $SA \overset{鈫�}{x} = S \overset{鈫�}{b}$ is $\overset{鈫�}{x} = [\begin{matrix} 7 \\ 11 \end{matrix}]$ .

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Most popular questions from this chapter

Find the angle $胃$ between each of the pairs of vectors $\overset{鈫�}{u}$ and in exercises 4 through 6.

5. $\overset{鈫�}{u} = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}| \overset{鈫�}{V} = [\begin{matrix} 2 \\ 3 \\ 4 \end{matrix}]$ .

Consider a symmetric $n 脳 m$ matrix A. What is the relationship between Im(A)and ker(A)?

Show that an orthogonal transformation Lfrom $R^{n}$ to $R^{n}$ preserves angles: The angle between two nonzero vectors $\overset{鈯�}{v}$ and $\overset{鈯�}{w}$ in $R^{n}$ equals the angle between $L (\overset{鈯�}{v})$ and $L (\overset{鈯�}{w})$ .Conversely, is any linear transformation that preserves angles orthogonal.

Is there an orthogonal transformation T from $R^{3}$ to $R^{3}$ such that $T [\begin{matrix} 2 \\ 3 \\ 0 \end{matrix}] = [\begin{matrix} 3 \\ 0 \\ 2 \end{matrix}]$

and $T [\begin{matrix} 2 \\ 3 \\ 0 \end{matrix}] = [\begin{matrix} 3 \\ 0 \\ 2 \end{matrix}]$ ?

Find the length of each of the vectors In exercises 1 through 3.

2. $\overset{鈬赌}{v} = [\begin{matrix} 2 \\ 3 \\ 4 \end{matrix}]$ .

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