Solve my Other example. Here is a photo

High School
Other

vinod singh

Answered question

2022-06-03

Answer & Explanation

xleb123

Skilled2023-05-19Added 181 answers

Given:

A = [\begin{matrix} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \end{matrix}]

b = [\begin{matrix} 3 \\ 5 \\ 7 \\ - 3 \end{matrix}]

To find the least-squares solution, we need to solve the normal equation

A^{T} A x = A^{T} b

, where

A^{T}

represents the transpose of matrix

A

.
1. Calculate

A^{T}

A^{T} = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 1 \end{matrix}]

2. Calculate

A^{T} A

A^{T} A = [\begin{matrix} 4 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 3 \end{matrix}]

3. Calculate

A^{T} b

A^{T} b = [\begin{matrix} 12 \\ 2 \\ 9 \end{matrix}]

4. Solve the normal equation

A^{T} A x = A^{T} b

[\begin{matrix} 4 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 3 \end{matrix}] x = [\begin{matrix} 12 \\ 2 \\ 9 \end{matrix}]

To solve this system of equations, we can use matrix inversion. Let

A^{T} A

be denoted as

C

C x = [\begin{matrix} 12 \\ 2 \\ 9 \end{matrix}]

x = C^{- 1} [\begin{matrix} 12 \\ 2 \\ 9 \end{matrix}]

To calculate the inverse of matrix

C

, we can use any method such as Gaussian elimination, LU decomposition, or matrix calculator.
After performing the necessary calculations, the solution for

x

is:

x = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]

Therefore, the least-squares solution of the system

A x = b

is:

x = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]

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