Q31E Fit a linear function of the for... [FREE SOLUTION]

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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: Q31E (page 247)

Fit a linear function of the form $f (t) = c_{0} + c_{1} t$ to the data points (0,3) , (1,3) , (1,6) using least square. Sketch the solution and explain why it make sense.

Short Answer

Expert verified

The linear function is $f (t) = 3 + \frac{1}{2} t$ .

Step by step solution

Step:1 Explanation of the solution

Consider the linear function as follows.

$f (t) = c_{0} + c_{1} t$

Substitute the points in the linear function as follows.

$(0, 3) 鈬� f (0) = c_{0} + c_{1} (0) = c_{0} = 3 (1, 3) 鈬� f (1) = c_{0} + c_{1} (1) = c_{0} + 3 c_{1} = 3 (1, 6) 鈬� f (1) = c_{0} + c_{1} (1) = c_{0} + 6 c_{1} = 6$

Then consider the system as $A \overset{鈫�}{x} = \overset{鈫�}{b}$ .

$where A = [\begin{matrix} 1 & 0 \\ 1 & 1 \\ 1 & 1 \end{matrix}], \overset{鈫�}{b} = [\begin{matrix} 3 \\ 3 \\ 6 \end{matrix}] and \overset{鈫�}{x} = [\begin{matrix} c_{0} \\ c_{1} \end{matrix}]$

To find the least square solution of the system as follows.

$A \overset{鈫�}{x} = \overset{鈫�}{b}$

The exact solution of the system is as follows.

$(A^{T} A) \overset{鈫�}{x} = A^{T} \overset{鈫�}{b} {[\begin{matrix} 1 & 0 \\ 1 & 0 \\ 1 & 1 \end{matrix}]}^{T} [\begin{matrix} 1 & 0 \\ 1 & 0 \\ 1 & 1 \end{matrix}] [\begin{matrix} c_{0} \\ c_{1} \end{matrix}] = {[\begin{matrix} 1 & 0 \\ 1 & 0 \\ 1 & 1 \end{matrix}]}^{T} [\begin{matrix} 3 \\ 6 \\ 6 \end{matrix}] [\begin{matrix} 1 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ 1 & 0 \\ 1 & 1 \end{matrix}] [\begin{matrix} c_{0} \\ c_{1} \end{matrix}] = [\begin{matrix} 1 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} 3 \\ 6 \\ 6 \end{matrix}] [\begin{matrix} 1 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ 1 & 0 \\ 1 & 1 \end{matrix}] [\begin{matrix} c_{0} \\ c_{1} \end{matrix}] = [\begin{matrix} 12 \\ 9 \end{matrix}]$

Simplify further as follows.

$[\begin{matrix} 3 & 2 \\ 2 & 2 \end{matrix}] [\begin{matrix} c_{0} \\ c_{1} \end{matrix}] = [\begin{matrix} 12 \\ 9 \end{matrix}] 3 c_{0} + 2 c_{1} = 12 c_{0} = 3, c_{1} = \frac{1}{2} [\begin{matrix} c_{0} \\ c_{1} \end{matrix}] = [\begin{matrix} 3 \\ \frac{1}{2} \end{matrix}] [\begin{matrix} c_{0} \\ c_{1} \end{matrix}] = [\begin{matrix} 3 \\ \frac{1}{2} \end{matrix}]$

Hence, the least square solution is $[\begin{matrix} c_{0} \\ c_{1} \end{matrix}] = [\begin{matrix} 3 \\ \frac{1}{2} \end{matrix}]$ .

Then the linear function is $f (t) = 3 + \frac{1}{2} t$

The graph for the linear function is as follows.

Hence, the linear function is $f (t) = 3 + \frac{1}{2} t$

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91影视

Fit a linear function of the form $f (t) = c_{0} + c_{1} t$ to the data points (0,3) , (1,3) , (1,6) using least square. Sketch the solution and explain why it make sense.

Short Answer

Step by step solution

Step:1 Explanation of the solution

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91影视

Fit a linear function of the form f(t)=c0+c1t to the data points (0,3) , (1,3) , (1,6) using least square. Sketch the solution and explain why it make sense.

Short Answer

Step by step solution

Step:1 Explanation of the solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Discrete Mathematics

Geometry

Applied Mathematics

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Fit a linear function of the form $f (t) = c_{0} + c_{1} t$ to the data points (0,3) , (1,3) , (1,6) using least square. Sketch the solution and explain why it make sense.