Find a basis for the solution space of the given homogeneous linear system. 5x4 X₁ - 3x₂ - 2X1 9x3- 3x2 12x3- X4 = 0 = X₁ + 2x₂ + X3 + 10x4 0 - = 0 =
Find a basis for the solution space of the given homogeneous linear system. 5x4 X₁ - 3x₂ - 2X1 9x3- 3x2 12x3- X4 = 0 = X₁ + 2x₂ + X3 + 10x4 0 - = 0 =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Find a basis for the solution space of the given homogeneous linear system.
**Equations:**
\[ x_1 - 3x_2 - 9x_3 - 5x_4 = 0 \]
\[ 2x_1 - 3x_2 - 12x_3 - x_4 = 0 \]
\[ x_1 + 2x_2 + x_3 + 10x_4 = 0 \]
**Discussion:**
This problem requires finding a set of vectors that form a basis for the solution space of the system, i.e., all solutions of the homogeneous system can be expressed as a linear combination of these basis vectors. Solving these equations will involve methods like Gaussian elimination or matrix row reduction to identify the free and leading variables and express the leading variables in terms of the free ones.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3200e892-7b83-4670-8aa5-c4c84f2a6adb%2Ff48e6640-c998-48e7-aedb-dd8025fb51c0%2Fmr64nok_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find a basis for the solution space of the given homogeneous linear system.
**Equations:**
\[ x_1 - 3x_2 - 9x_3 - 5x_4 = 0 \]
\[ 2x_1 - 3x_2 - 12x_3 - x_4 = 0 \]
\[ x_1 + 2x_2 + x_3 + 10x_4 = 0 \]
**Discussion:**
This problem requires finding a set of vectors that form a basis for the solution space of the system, i.e., all solutions of the homogeneous system can be expressed as a linear combination of these basis vectors. Solving these equations will involve methods like Gaussian elimination or matrix row reduction to identify the free and leading variables and express the leading variables in terms of the free ones.
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