Math Problem Statement

solve pls

Investigate whether a homogeneous system of equations has non-trivial solutions. In case of a positive answer, find its general solution. Also, write down the fundamental system of solutions.

Solution

We are given the following system of linear equations:

1.\quad x_1 + 2x_2 + 4x_3 - 3x_4 &= 0, \\ 2.\quad 3x_1 + 5x_2 + 6x_3 - 4x_4 &= 0, \\ 3.\quad 4x_1 + 5x_2 - 2x_3 + 3x_4 &= 0, \\ 4.\quad 3x_1 + 8x_2 + 24x_3 - 19x_4 &= 0. \end{aligned}$$ ### Step 1: Convert the system to an augmented matrix The augmented matrix of the system is: $$\begin{bmatrix} 1 & 2 & 4 & -3 & 0 \\ 3 & 5 & 6 & -4 & 0 \\ 4 & 5 & -2 & 3 & 0 \\ 3 & 8 & 24 & -19 & 0 \end{bmatrix}$$ ### Step 2: Perform Gaussian elimination to row-reduce the matrix I will now calculate the row-reduced echelon form (RREF) of this matrix to determine the solution. Let's proceed with that computation.The row-reduced echelon form (RREF) of the augmented matrix is: $$\begin{bmatrix} 1 & 0 & -8 & 7 & 0 \\ 0 & 1 & 6 & -5 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$ ### Step 3: Interpret the result The system has two pivot columns (column 1 and column 2), which correspond to the variables $$x_1$$ and $$x_2$$. The third and fourth columns, corresponding to $$x_3$$ and $$x_4$$, are free variables. Therefore, the system has infinitely many solutions, and non-trivial solutions exist. ### Step 4: General solution From the RREF matrix, we can express the leading variables $$x_1$$ and $$x_2$$ in terms of the free variables $$x_3$$ and $$x_4$$: $$\begin{aligned} x_1 &= 8x_3 - 7x_4, \\ x_2 &= -6x_3 + 5x_4. \end{aligned}$$ Thus, the general solution is: $$\begin{aligned} x_1 &= 8x_3 - 7x_4, \\ x_2 &= -6x_3 + 5x_4, \\ x_3 &= x_3, \\ x_4 &= x_4. \end{aligned}$$ ### Step 5: Fundamental system of solutions To find the fundamental system of solutions, we express the general solution in terms of the free variables $$x_3$$ and $$x_4$$. Setting $$x_3 = 1, x_4 = 0$$ and then $$x_3 = 0, x_4 = 1$$, we get the following two independent solutions: 1. When $$x_3 = 1, x_4 = 0$$: $$(x_1, x_2, x_3, x_4) = (8, -6, 1, 0)$$ 2. When $$x_3 = 0, x_4 = 1$$: $$(x_1, x_2, x_3, x_4) = (-7, 5, 0, 1)$$ Therefore, the fundamental system of solutions is: $$\{ (8, -6, 1, 0), (-7, 5, 0, 1) \}$$ Would you like further clarification or details on any step? ### 5 Related Questions: 1. What is the significance of free variables in a system of linear equations? 2. How can we determine if a homogeneous system has non-trivial solutions? 3. What is the geometric interpretation of the solution space for this system? 4. How would this solution change if one of the equations was different? 5. Can the rank of a matrix always predict the number of free variables? **Tip:** When solving homogeneous systems, always check the rank of the coefficient matrix compared to the number of variables to determine the number of free variables.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Homogeneous Systems of Equations
Gaussian Elimination

Formulas

Augmented matrix representation
Row-reduced echelon form (RREF)

Theorems

Fundamental Theorem of Linear Systems
Rank-Nullity Theorem

Suitable Grade Level

University level (Linear Algebra)

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