Math Problem Statement

Solve the homogeneous linear system corresponding to the given coefficient matrix.

Solution

Let's analyze the system of linear equations based on the matrix provided.

The matrix is:

1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ This corresponds to the following system of homogeneous linear equations (since it's homogeneous, the right-hand side is all zeros): 1. $$x_1 + x_4 = 0$$ 2. $$x_3 = 0$$ 3. No equation for the third row (as it is entirely zero). Now, let's solve this: ### From the first equation: $$x_1 + x_4 = 0$$ $$\Rightarrow x_1 = -x_4$$ ### From the second equation: $$x_3 = 0$$ ### Free variables: - $$x_2$$ and $$x_4$$ are free variables because they are not restricted by any equations. ### General solution: We can express $$x_1, x_2, x_3, x_4$$ in terms of the free variables $$x_2$$ and $$x_4$$, using parameters $$t$$ and $$s$$ respectively. Let: - $$x_2 = t$$ - $$x_4 = s$$ Thus, the general solution is: $$x_1 = -s, \quad x_2 = t, \quad x_3 = 0, \quad x_4 = s$$ In vector form, the solution is: $$(x_1, x_2, x_3, x_4) = (-s, t, 0, s)$$ ### Final answer: $$(x_1, x_2, x_3, x_4) = (-s, t, 0, s)$$ Would you like further details or have any other questions? Here are some related questions to deepen your understanding: 1. What is a homogeneous system of linear equations? 2. How do you determine free variables in a linear system? 3. What does it mean for a system to have an infinite number of solutions? 4. How would the solution change if the system was non-homogeneous? 5. How can matrix row reduction help solve systems of linear equations? **Tip:** In a homogeneous system, if there are free variables, it implies there are infinitely many solutions!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Homogeneous Systems
Matrix Equations

Formulas

Ax = 0
General solution for free variables

Theorems

Existence and Uniqueness Theorem for Linear Systems
Linear Independence

Suitable Grade Level

Undergraduate

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