Q26E Find the least square聽x鈬赌* of ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q26E (page 247)

Find the least square ${\overset{鈬赌}{x}}^{*}$ of the system $A \overset{鈬赌}{x} = b$ where $A = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}]$ and $\overset{鈬赌}{b} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]$ .

Short Answer

Expert verified

The solution $\overset{鈬赌}{x} = [\begin{matrix} - \frac{7}{6} \\ 1 \\ 0 \end{matrix}] + t [\begin{matrix} 1 \\ - 2 \\ 1 \end{matrix}]$ .

Step by step solution

Explanation of the solution

Consider the system as follows.

$A \overset{鈬赌}{x} = b$ where $A = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}]$ and $\overset{鈬赌}{b} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]$ .

Consider the matrix A as follows.

$A = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}]$

Now, find $A^{T} A$ as follows.

$A^{T} A = {[\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}]}^{T} [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}] = [\begin{matrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{matrix}] [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}] = [\begin{matrix} 66 & 78 & 90 \\ 78 & 93 & 108 \\ 90 & 108 & 126 \end{matrix}] A^{T} A = [\begin{matrix} 66 & 78 & 90 \\ 78 & 93 & 108 \\ 90 & 108 & 126 \end{matrix}]$

The exact solution of the system is as follows.

$(A^{T} A) \overset{鈬赌}{x} = A^{T} \overset{鈬赌}{b} [\begin{matrix} 66 & 78 & 90 \\ 78 & 93 & 108 \\ 90 & 108 & 126 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}] [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] [\begin{matrix} 66 & 78 & 90 \\ 78 & 93 & 108 \\ 90 & 108 & 126 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] [\begin{matrix} 66 & 78 & 90 \\ 78 & 93 & 108 \\ 90 & 108 & 126 \end{matrix}] [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$

After solving the matrix calculation as follows.

$[\begin{matrix} 1 & 0 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{matrix} \begin{matrix} - \frac{7}{6} \\ 1 \\ 0 \end{matrix}]$

Then write the equation as follows.

$x_{1} - x_{3} = - \frac{7}{6} x_{2} + 2 x_{3} = 1 x_{2} + 2 x_{3} = 1 x_{3} = t$

Simplify further as follows.

$[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} - \frac{7}{6} + t \\ 1 - 2 t \\ t \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] [\begin{matrix} - \frac{7}{6} + t \\ 1 \\ 0 \end{matrix}] + t [\begin{matrix} 1 \\ - 2 \\ 3 \end{matrix}]$

Hence, the least square for the system $(A^{T} A) \overset{鈫�}{x} = A^{T} \overset{鈬赌}{b}$ is $\overset{鈬赌}{x} = [\begin{matrix} - \frac{7}{6} \\ 1 \\ 0 \end{matrix}] + t [\begin{matrix} 1 \\ - 2 \\ 1 \end{matrix}]$ .

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91影视

Find the least square ${\overset{鈬赌}{x}}^{*}$ of the system $A \overset{鈬赌}{x} = b$ where $A = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}]$ and $\overset{鈬赌}{b} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]$ .

Short Answer

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91影视

Find the least squarex鈬赌* of the systemAx鈬赌=b whereA=[123456789] andb鈬赌=[100] .

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Find the least square ${\overset{鈬赌}{x}}^{*}$ of the system $A \overset{鈬赌}{x} = b$ where $A = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}]$ and $\overset{鈬赌}{b} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]$ .