Describe the set of solutions x=(x1, x2, x3) ∈ℝ^3 of the system of equations . x1-x2+x3 =

step 1 The question asks to find the solution of the given system. First, we need to change the system

Describe the set of solutions x=(x1, x2, x3) ∈ℝ^3 of the...

Describe the set of solutions $x=\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}$ of the system of equations
$$\left.\begin{array}{rl}
x_{1}-x_{2}+x_{3} & =0 \\x_{1}+2 x_{2}+x_{3} & =0 \\
2 x_{1}+x_{2}+2 x_{3} & =0\end{array}\right\}$$

Key Concepts

Parameterization of the Solution Set

When the number of variables exceeds the rank of the coefficient matrix in a homogeneous system, there are free variables that can be assigned arbitrary values. The solution set is then expressed in terms of these parameters, effectively creating a parameterized description that captures the infinite set of solutions as a vector subspace. This method is pivotal in providing a complete description of the solution space.

Null Space of a Matrix

The null space (or kernel) of a matrix is the set of all vectors that, when multiplied by the matrix, yield the zero vector. In the context of a homogeneous linear system, the set of solutions exactly coincides with the null space of the coefficient matrix. Understanding the null space is essential for describing the solution set as it provides insights into the dimensionality and structure of all possible solutions.

Homogeneous Linear Systems

A homogeneous system of linear equations is one in which all of the constant terms are zero. Such systems always have at least the trivial solution (where all variables are zero), and they are particularly important because their solution sets form a vector subspace. This concept underlies many applications in linear algebra, especially in understanding the structure of solutions and the concept of linear dependence.

Matrix Representation and Row Reduction

Systems of linear equations are often represented in matrix form, which allows systematic methods such as row reduction (or Gaussian elimination) to be applied. Row reduction transforms the matrix into a simpler form (typically row echelon or reduced row echelon form) to easily identify dependent equations, free variables, and the overall structure of the solution set.

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