Math Problem Statement

Solve the homogeneous linear system corresponding to the given coefficient matrix for x₁, x₂, x₃, and x₄ in terms of the parameters t and s.

Solution

The given problem asks to solve the homogeneous linear system corresponding to the matrix:

1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ The solution is requested in terms of parameters $$t$$ and $$s$$. ### Step-by-step Solution: This is a homogeneous system, meaning the system is of the form $$A \mathbf{x} = 0$$, where $$A$$ is the given coefficient matrix and $$\mathbf{x} = (x_1, x_2, x_3, x_4)$$. To solve this: 1. **Convert the system to equations**: - From row 1: $$x_1 + x_4 = 0$$ $$\Rightarrow x_1 = -x_4$$ - From row 2: $$x_2 + x_4 = 0$$ $$\Rightarrow x_2 = -x_4$$ - From row 3: $$x_3 = 0$$ - Row 4 does not give any new information. 2. **Parameterize the free variables**: - Let $$x_4 = t$$, where $$t$$ is a free parameter. 3. **Write the solution in terms of $$t$$**: - From row 1 and row 2: $$x_1 = -t$$ and $$x_2 = -t$$. - From row 3: $$x_3 = 0$$. - $$x_4 = t$$. Thus, the solution vector can be written as: $$(x_1, x_2, x_3, x_4) = (-t, -t, 0, t)$$ This is the general solution for the system in terms of $$t$$. Would you like more details on any specific part of the solution? --- Here are five related questions to deepen your understanding: 1. How would the solution change if a non-homogeneous term were added to the system? 2. What happens if there are multiple free variables in the system? 3. How do we interpret a row of all zeros in a matrix like this one? 4. How does Gaussian elimination work for solving systems like this? 5. How can you check if a system has infinitely many solutions? **Tip:** When solving homogeneous systems, always identify free variables to express the general solution in terms of parameters!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Homogeneous Systems
Gaussian Elimination

Formulas

Ax = 0
General solution of a system in terms of free variables

Theorems

Rank-Nullity Theorem
Gaussian Elimination

Suitable Grade Level

College-level (Intro to Linear Algebra)

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