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1 30-1 0 4 Given the matrix A is row equivalent to 0 0 1 -4 0-2 describe all solutions to Ax = 0. 0 0 0 0 1 X = X2 + X4 + X6 > Next Question

1 30-1 0 4 Given the matrix A is row equivalent to 0 0 1 -4 0-2 describe all solutions to Ax = 0. 0 0 0 0 1 X = X2 + X4 + X6 > Next Question

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Describing Solutions to the Linear System \(Ax = 0\) Given the matrix \(A\) is row equivalent to \[ \begin{bmatrix} 1 & 3 & 0 & -1 & 0 & 4 \\ 0 & 0 & 1 & 4 & 0 & -2 \\ 0 & 0 & 0 & 0 & 1 & 5 \end{bmatrix}, \] describe all solutions to \(Ax = 0\). \[ \mathbf{x} = \begin{bmatrix} \\ \\ \\ \\ \\ \\ \\ \end{bmatrix} = x_2 \begin{bmatrix} \\ \\ \\ \\ \\ \\ \\ \end{bmatrix} + x_4 \begin{bmatrix} \\ \\ \\ \\ \\ \\ \\ \end{bmatrix} + x_6 \begin{bmatrix} \\ \\ \\ \\ \\ \\ \\ \end{bmatrix} \] This matrix represents a system of linear equations. To describe all solutions to the equation \(Ax = 0\), you need to express the variable vector \(\mathbf{x}\) in terms of the basic variables and the free variables. Here, columns with no leading 1s in the row-reduced echelon form (columns 2, 4, and 6) correspond to free variables \(x_2\), \(x_4\), and \(x_6\). In the image, there are placeholders for filling in the expressions for the free variables \(x_2, x_4,\) and \(x_6\) which combine to describe the general solution to the system. The solutions are vectors that will be combinations of the free variables. Beyond this explanation, students will typically need to solve the system to express \(\mathbf{x}\) explicitly in terms of \(x_2, x_4,\) and \(x_6\).
Transcribed Image Text:### Describing Solutions to the Linear System \(Ax = 0\) Given the matrix \(A\) is row equivalent to \[ \begin{bmatrix} 1 & 3 & 0 & -1 & 0 & 4 \\ 0 & 0 & 1 & 4 & 0 & -2 \\ 0 & 0 & 0 & 0 & 1 & 5 \end{bmatrix}, \] describe all solutions to \(Ax = 0\). \[ \mathbf{x} = \begin{bmatrix} \\ \\ \\ \\ \\ \\ \\ \end{bmatrix} = x_2 \begin{bmatrix} \\ \\ \\ \\ \\ \\ \\ \end{bmatrix} + x_4 \begin{bmatrix} \\ \\ \\ \\ \\ \\ \\ \end{bmatrix} + x_6 \begin{bmatrix} \\ \\ \\ \\ \\ \\ \\ \end{bmatrix} \] This matrix represents a system of linear equations. To describe all solutions to the equation \(Ax = 0\), you need to express the variable vector \(\mathbf{x}\) in terms of the basic variables and the free variables. Here, columns with no leading 1s in the row-reduced echelon form (columns 2, 4, and 6) correspond to free variables \(x_2\), \(x_4\), and \(x_6\). In the image, there are placeholders for filling in the expressions for the free variables \(x_2, x_4,\) and \(x_6\) which combine to describe the general solution to the system. The solutions are vectors that will be combinations of the free variables. Beyond this explanation, students will typically need to solve the system to express \(\mathbf{x}\) explicitly in terms of \(x_2, x_4,\) and \(x_6\).
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