Math Problem Statement
Let A = [[2, 0, 6], [-1, 8, 5], [1, -2, 1]] and b = [10, 3, 3], and let W be the set of all linear combinations of the columns of A. Determine if b is in W.
Solution
Problem Analysis:
We are given the matrix , and the vector .
The set is defined as the span of the columns of , i.e., the set of all linear combinations of the columns of . The goal is to determine if is in .
This is equivalent to solving the matrix equation: where are the coefficients of the linear combination of the columns of . If a solution exists, then is in ; otherwise, it is not.
Solving the System:
We solve :
\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 10 \\ 3 \\ 3 \end{bmatrix}.$$ This can be solved by writing the augmented matrix and performing row reduction. $$\text{Augmented matrix:} \quad \left[\begin{array}{ccc|c} 2 & 0 & 6 & 10 \\ -1 & 8 & 5 & 3 \\ 1 & -2 & 1 & 3 \end{array}\right].$$ --- Let me compute the solution step-by-step. numpy as np # Define matrix A and vector b A = np.array([[2, 0, 6], [-1, 8, 5], [1, -2, 1]]) b = np.array([10, 3, 3]) # Solve the augmented matrix using numpy.linalg.lstsq to determine if a solution exists solution, residuals, rank, singular_values = np.linalg.lstsq(A, b, rcond=None) # Check if the solution satisfies the equation A*x = b (or if b is in the column space of A) b_is_in_span = np.allclose(np.dot(A, solution), b) solution, residuals, rank, singular_values, b_is_in_span### Solution: The system $$A\mathbf{x} = \mathbf{b}$$ has a solution. The computed solution is: $$\mathbf{x} = \begin{bmatrix} 0.6364 \\ -0.4545 \\ 1.4545 \end{bmatrix}.$$ The rank of $$A$$ is 2, and the vector $$\mathbf{b}$$ lies within the column space (span of the columns) of $$A$$. Therefore, $$\mathbf{b} \in W$$. --- ### Conclusion: Since $$\mathbf{b}$$ is in the span of the columns of $$A$$, we conclude that: $$\mathbf{b} \in W.$$ Would you like a step-by-step explanation of the row reduction, or more details on the solution process? Let me know! --- ### Related Questions: 1. What does it mean geometrically for $$\mathbf{b}$$ to be in the span of the columns of $$A$$? 2. How would the solution change if the rank of $$A$$ was less than 3? 3. How is the rank of $$A$$ related to the existence of solutions? 4. What happens if $$\mathbf{b}$$ is not in $$W$$? Can we still compute a "least-squares" solution? 5. How can we verify computationally that $$\mathbf{b}$$ is in $$W$$? ### Tip: When solving systems of linear equations, checking the rank of the matrix helps you understand whether the system is consistent (has a solution) or inconsistent (no solution).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Equations
Span of Vectors
Formulas
A * x = b
Row Reduction of Augmented Matrices
Theorems
Span of a Set of Vectors
Rank-Nullity Theorem
Suitable Grade Level
College Level (Undergraduate)