Math Problem Statement
Let bequalsStart 4 By 1 Table 1st Row 1st Column 6 2nd Row 1st Column negative 4 3rd Row 1st Column 26 4st Row 1st Column 2 EndTable and let AequalsStart 4 By 4 Table 1st Row 1st Column 5 2nd Column negative 3 3rd Column 4 4st Column negative 2 2nd Row 1st Column negative 7 2nd Column 5 3rd Column negative 6 4st Column 0 3rd Row 1st Column negative 5 2nd Column 7 3rd Column 4 4st Column negative 12 4st Row 1st Column 8 2nd Column negative 6 3rd Column 6 4st Column 1 EndTable . Is b in the range of the transformation Bold x maps to Upper A Bold x? If so, find an x whose image under the transformation is b. Question content area bottom Part 1 Is b in the range of the transformation? A. No, because the appropriate augmented matrix shows an inconsistent system. B. Yes, because the appropriate augmented matrix shows an inconsistent system. C. Yes, because the appropriate augmented matrix shows a consistent system. Your answer is correct.D. No, because the appropriate augmented matrix shows a consistent system. Part 2 Find an x whose image under the transformation is b. xequals
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Solution
To determine if b is in the range of the transformation x → Ax, we need to check if the system Ax = b has a solution. This can be done by forming the augmented matrix [A | b] and row reducing it to check for consistency.
Step 1: Form the Augmented Matrix
We construct the augmented matrix [A | b]:
5 & -3 & 4 & -2 & | 6 \\ -7 & 5 & -6 & 0 & | -4 \\ -5 & 7 & 4 & -12 & | 26 \\ 8 & -6 & 6 & 1 & | 2 \end{bmatrix}$$ We row-reduce this matrix to check for consistency. --- ### **Step 2: Row Reduction** Using Gaussian elimination, we perform row operations: 1. Make the first pivot 1 by dividing the first row by 5. 2. Use row operations to make the first column below the pivot **0**. 3. Continue eliminating entries to achieve row echelon form. 4. Check if the system is consistent (i.e., does not lead to a contradiction like **0 = 1**). After row reduction, we find that the system is **consistent** (i.e., there is no row leading to an equation like **0 = 1**). This confirms that **b is in the range of the transformation**. Thus, the correct answer to **Part 1** is: ✅ **C. Yes, because the appropriate augmented matrix shows a consistent system.** --- ### **Step 3: Solve for x** Now, solving the reduced system for **x**, we obtain: $$x = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}$$ Thus, the solution to **Ax = b** is: $$\mathbf{x} = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}$$ --- ### **Final Answer:** 1. **b is in the range of the transformation** ✅ **(Answer: C - The system is consistent)** 2. **Solution for x:** $$x = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}$$ --- Would you like a step-by-step breakdown of the row reduction? Here are five related questions to explore further: 1. How do you determine the range of a transformation mathematically? 2. What does it mean for a system to be inconsistent? 3. How do you compute the null space of a matrix? 4. What is the significance of the rank of a matrix in linear transformations? 5. How does Gaussian elimination help in solving systems of equations? **Tip:** If the augmented matrix has a row like **[0 0 0 0 | k]** (where **k ≠ 0**), the system is inconsistent.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Operations
Gaussian Elimination
Range of a Transformation
Formulas
Ax = b
Theorems
Range of a Linear Transformation
Gaussian Elimination Theorem
Suitable Grade Level
Grades 11-12
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