Math Problem Statement

please solve

The following system of linear equations is inconsistent: x + y - z = 7, x + 2y - 7z = -2, 2x + 3y - 8z = -7, 3x + 2y + 3z = 3. Find any x, y, z that gives the 'closest' possible answer.

Solution

The system of linear equations is inconsistent, meaning there is no exact solution where all equations are simultaneously satisfied. Instead, we need to find a "closest" solution, often interpreted as the least-squares solution.

The given system is:

  1. x+y−z=7x + y - z = 7
  2. x+2y−7z=−2x + 2y - 7z = -2
  3. 2x+3y−8z=−72x + 3y - 8z = -7
  4. 3x+2y+3z=33x + 2y + 3z = 3

Approach:

We write the system in matrix form Av=bA \mathbf{v} = \mathbf{b}, where:

\begin{bmatrix} 1 & 1 & -1 \\ 1 & 2 & -7 \\ 2 & 3 & -8 \\ 3 & 2 & 3 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 7 \\ -2 \\ -7 \\ 3 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}.$$ Since the system is inconsistent, we solve using the **least-squares method**, which minimizes $$\| A\mathbf{v} - \mathbf{b} \|^2$$. The least-squares solution is given by: $$\mathbf{v} = (A^\top A)^{-1} A^\top \mathbf{b}.$$ Let me compute this. numpy as np # Define the matrix A and vector b A = np.array([ [1, 1, -1], [1, 2, -7], [2, 3, -8], [3, 2, 3] ]) b = np.array([7, -2, -7, 3]) # Compute the least-squares solution: v = (A^T A)^(-1) A^T b AtA = A.T @ A Atb = A.T @ b v = np.linalg.inv(AtA) @ Atb vThe matrix $$A^T A$$ is singular, so its inverse cannot be computed directly. Instead, we can use a **pseudo-inverse** to find the least-squares solution. Let me compute this using the Moore-Penrose pseudo-inverse. Compute the least-squares solution using the pseudo-inverse v_pseudo = np.linalg.pinv(A) @ b v_pseudoThe least-squares solution is approximately: $$x = 0.452, \quad y = 0.258, \quad z = 0.710$$ This provides the "closest" possible solution to the inconsistent system. Would you like details on how the least-squares method works? Here are 5 related questions you could explore: 1. How does the least-squares method minimize error in inconsistent systems? 2. Can we interpret the pseudo-inverse geometrically? 3. How does matrix singularity affect solving linear systems? 4. How do small changes in $$A$$ or $$b$$ affect the least-squares solution? 5. What is the practical significance of least-squares in real-world applications? **Tip:** For better accuracy, ensure all equations are scaled similarly before applying least-squares.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Operations
Least-Squares Approximation
Pseudo-Inverse

Formulas

Matrix form: A*v = b
Least-squares solution: v = (A^T * A)^(-1) * A^T * b
Pseudo-inverse solution: v = pinv(A) * b

Theorems

Least-Squares Method
Moore-Penrose Pseudo-Inverse

Suitable Grade Level

Undergraduate (Linear Algebra course or equivalent)

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