Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q25E (page 247)

Find the least-squares solution $\overset{鈫�}{x}$ of the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ , where $A = [\begin{matrix} 1 & 3 \\ 2 & 6 \end{matrix}]$ and $\overset{鈫�}{b} = [\begin{matrix} 5 \\ 6 \end{matrix}]$ . Use only paper and pencil. Draw a sketch.

Short Answer

Expert verified

The least square solution of ${\overset{鈫�}{x}}^{*}$ is $\overset{鈫�}{x_{1}} = 1 - 3 \overset{鈫�}{x_{2}}$ and the graph is

Step by step solution

Determine the least squares solution.

Consider the solution of the equation matrix $A \overset{鈫�}{x} = \overset{鈫�}{b}$ where $A = [\begin{matrix} 1 & 3 \\ 2 & 6 \end{matrix}]$ and $\overset{鈫�}{b} = [\begin{matrix} 5 \\ 0 \end{matrix}]$ .

If $\overset{鈫�}{x}$ is the solution of the equation $A \overset{鈫�}{x} = \overset{鈫�}{b}$ then the least-square solution ${\overset{鈫�}{x}}^{*}$ is defined as $(A^{T} A) {\overset{鈫�}{x}}^{*} = A^{T} \overset{鈫�}{b}$ .

Substitute the value $[\begin{matrix} 1 & 3 \\ 2 & 6 \end{matrix}]$ for A and $[\begin{matrix} 3 \\ 2 \\ 7 \end{matrix}]$ for $\overset{鈫�}{b}$ in the equation $(A^{T} A) {\overset{鈫�}{x}}^{*} = A^{T} \overset{鈫�}{b}$ .

$(A^{T} A) {\overset{鈫�}{x}}^{*} A^{T} \overset{鈫�}{b} ({[\begin{matrix} 1 & 3 \\ 2 & 6 \end{matrix}]}^{T} [\begin{matrix} 1 & 3 \\ 2 & 6 \end{matrix}]) {\overset{鈫�}{x}}^{*} = {[\begin{matrix} 1 & 3 \\ 2 & 6 \end{matrix}]}^{T} [\begin{matrix} 5 \\ 0 \end{matrix}] ([\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}] [\begin{matrix} 1 & 3 \\ 2 & 6 \end{matrix}]) {\overset{鈫�}{x}}^{*} = {[\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}]}^{T} [\begin{matrix} 5 \\ 0 \end{matrix}] ([\begin{matrix} 5 & 15 \\ 15 & 45 \end{matrix}]) {\overset{鈫�}{x}}^{*} = [\begin{matrix} 5 \\ 15 \end{matrix}]$

Further, simplify the equation as follows.

$([\begin{matrix} 5 & 15 \\ 15 & 45 \end{matrix}]) [\begin{matrix} {\overset{鈫�}{x}}_{1} \\ {\overset{鈫�}{x}}_{2} \end{matrix}] = [\begin{matrix} 5 \\ 15 \end{matrix}] 5 {\overset{鈫�}{x}}_{1} = 15 {\overset{鈫�}{x}}_{2} = 5 {\overset{鈫�}{x}}_{1} = 1 - 3 {\overset{鈫�}{x}}_{2}$

Therefore, the least square solution of ${\overset{鈫�}{x}}^{*}$ is in the form of ${\overset{鈫�}{x}}_{1} = 1 - 3 {\overset{鈫�}{x}}_{2}$ .

Draw the graph of.

Draw the graph of the least square solution of the equation $\overset{鈫�}{x_{1}} = 1 - 3 \overset{鈫�}{x_{2}}$ as follows.

Hence, the least square solution of ${\overset{鈫�}{x}}^{*}$ is ${\overset{鈫�}{x}}_{1} = 1 - 3 {\overset{鈫�}{x}}_{2}$ and the graph is sketched.

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91影视

Find the least-squares solution $\overset{鈫�}{x}$ of the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ , where $A = [\begin{matrix} 1 & 3 \\ 2 & 6 \end{matrix}]$ and $\overset{鈫�}{b} = [\begin{matrix} 5 \\ 6 \end{matrix}]$ . Use only paper and pencil. Draw a sketch.

Short Answer

Step by step solution

Determine the least squares solution.

Draw the graph of.

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91影视

Find the least-squares solutionx鈫� of the systemAx鈫�=b鈫�, whereA=[1326] andb鈫�=[56]. Use only paper and pencil. Draw a sketch.

Short Answer

Step by step solution

Determine the least squares solution.

Draw the graph of.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Applied Mathematics

Mechanics Maths

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Find the least-squares solution $\overset{鈫�}{x}$ of the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ , where $A = [\begin{matrix} 1 & 3 \\ 2 & 6 \end{matrix}]$ and $\overset{鈫�}{b} = [\begin{matrix} 5 \\ 6 \end{matrix}]$ . Use only paper and pencil. Draw a sketch.