Math Problem Statement

Describe all solutions of Axequals0 in parametric vector​ form, where A is row equivalent to the given matrix. Start 2 By 4 Table 1st Row 1st Column 1 2nd Column 4 3rd Column 0 4st Column negative 2 2nd Row 1st Column 2 2nd Column 8 3rd Column 0 4st Column negative 4 EndTable

Question content area bottom Part 1 xequalsx Subscript 2 select: v@Sub{2} enter your response hereplusx Subscript 3    enter your response hereplusx Subscript 4    enter your response here ​(Type an integer or fraction for each matrix​ element.)

Solution

We are given a matrix AA that is row equivalent to the following:

1 & 4 & 0 & -2 \\ 2 & 8 & 0 & -4 \\ \end{pmatrix}$$ We need to solve the homogeneous system $$A\mathbf{x} = 0$$, where $$\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix}$$, and express the solution in parametric vector form. ### Step 1: Row Reduction First, we reduce the given matrix $$A$$ to row echelon form. Start with the matrix: $$\begin{pmatrix} 1 & 4 & 0 & -2 \\ 2 & 8 & 0 & -4 \\ \end{pmatrix}$$ Perform the row operation $$R_2 \rightarrow R_2 - 2R_1$$: $$\begin{pmatrix} 1 & 4 & 0 & -2 \\ 0 & 0 & 0 & 0 \\ \end{pmatrix}$$ The second row is now entirely zeros, meaning we have one non-trivial equation. ### Step 2: Solve the System The system of equations derived from the first row is: $$x_1 + 4x_2 - 2x_4 = 0$$ Solving for $$x_1$$ in terms of the free variables $$x_2, x_3,$$ and $$x_4$$: $$x_1 = -4x_2 + 2x_4$$ Since $$x_2, x_3,$$ and $$x_4$$ are free variables, we can express the solution in parametric form as follows: $$\mathbf{x} = x_2 \begin{pmatrix} -4 \\ 1 \\ 0 \\ 0 \end{pmatrix} + x_3 \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} + x_4 \begin{pmatrix} 2 \\ 0 \\ 0 \\ 1 \end{pmatrix}$$ ### Final Solution: The general solution in parametric vector form is: $$\mathbf{x} = x_2 \begin{pmatrix} -4 \\ 1 \\ 0 \\ 0 \end{pmatrix} + x_3 \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} + x_4 \begin{pmatrix} 2 \\ 0 \\ 0 \\ 1 \end{pmatrix}$$ ### Summary: - $$x_1 = -4x_2 + 2x_4$$ - $$x_2, x_3,$$ and $$x_4$$ are free variables. Let me know if you need further clarification, or if you'd like to discuss any part of the process. --- Here are five questions to deepen your understanding: 1. What is the geometric interpretation of the parametric vector solution? 2. How would the solution change if the matrix $$A$$ had three rows instead of two? 3. What does it mean for a variable to be "free" in the context of linear algebra? 4. Could the system $$A\mathbf{x} = 0$$ have more than one solution form if the matrix were different? 5. How does row reduction help in solving systems of linear equations? **Tip:** When solving homogeneous systems, always check the rank of the matrix, as it helps in identifying the number of free variables.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Homogeneous Systems
Parametric Vector Form
Row Reduction

Formulas

Row reduction and echelon form of a matrix
General solution of Ax = 0 in parametric vector form

Theorems

Row Echelon Form Theorem
Rank-Nullity Theorem

Suitable Grade Level

College Level - Linear Algebra

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