Math Problem Statement
Let A=[4,-5,-3 5,-2,6. -4, 5, 4. 8, -27, -47] and b=[-2 -47 -3 184] Define the transformation T: R^3 -> R^4 by T(x) = Ax . Find a vector x whose image under T is b .
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x=[?,?,?]
Solution
To find a vector such that , we need to solve the system of equations represented by .
Given:
4 & -5 & -3 \\ 5 & -2 & 6 \\ -4 & 5 & 4 \\ 8 & -27 & -47 \end{bmatrix}, \quad b = \begin{bmatrix} -2 \\ -47 \\ -3 \\ 184 \end{bmatrix}$$ We need to solve the matrix equation: $$\begin{bmatrix} 4 & -5 & -3 \\ 5 & -2 & 6 \\ -4 & 5 & 4 \\ 8 & -27 & -47 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -2 \\ -47 \\ -3 \\ 184 \end{bmatrix}$$ This system of linear equations can be solved using Gaussian elimination, or alternatively through matrix methods such as LU decomposition or using a computer algorithm. Let's proceed with solving it.The solution to the system $$Ax = b$$ is: $$x = \begin{bmatrix} -3 \\ 1 \\ -5 \end{bmatrix}$$ Thus, the vector $$x = [-3, 1, -5]$$ is the one whose image under the transformation $$T(x) = Ax$$ is $$b$$. Do you want more details on the solution process or have any questions? Here are 5 related questions for further exploration: 1. How do we know that the solution for $$x$$ is unique? 2. What is the geometric interpretation of the transformation $$T$$? 3. How can we verify the result by back-substitution? 4. What other methods can be used to solve this system? 5. How does the rank of matrix $$A$$ affect the solution? **Tip:** When solving a system of linear equations, using matrix decomposition methods (like LU or QR) can sometimes be more efficient than direct Gaussian elimination, especially for large matrices.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Transformations
Systems of Linear Equations
Formulas
Matrix equation Ax = b
Gaussian Elimination
LU Decomposition
Theorems
Fundamental Theorem of Linear Algebra
Uniqueness of solutions for full-rank matrices
Suitable Grade Level
College Level (Linear Algebra)
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