91Ó°ÊÓ

The foundation of finding the nullspace of a matrix is setting up the matrix equation. This process involves stating the equation involving matrix multiplication that results in a zero vector. For the matrix \(A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}\), we are tasked with determining the vector \(\mathbf{x}\) such that \(A \mathbf{x} = \mathbf{0}\). Here, \(\mathbf{x}\) is a column vector, \(\begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}\), and \(\mathbf{0}\) is the zero vector, \(\begin{bmatrix} 0 \end{bmatrix}\). This lays the groundwork for seeking the nullspace by establishing a relationship between the given matrix and the unknown vector. It's essential to understand that this equation represents a system of linear equations. In this case, one equation is derived: \[ x_1 + 2x_2 + 3x_3 = 0 \]. By solving this equation, we dive into the realm of vector solutions.

Once the matrix equation is established, the next step is to find solutions for the vector \(\mathbf{x}\) that satisfy the equation. Since in our case, the matrix \(A\) is a single-row matrix leading to a single equation like \(x_1 + 2x_2 + 3x_3 = 0\), it presents us with infinite possibilities for solutions, given there are more variables than equations. To find these vector solutions, we express one variable in terms of others. In particular, solving for \(x_1\) in terms of \(x_2\) and \(x_3\) gives us: - \(x_1 = -2x_2 - 3x_3\)This step reveals that every solution for this system can be expressed as a linear combination of vectors, indicating the infinite possibilities contained within vector solutions. By utilizing parameters, we can succinctly capture these myriad solutions through a compact representation.

Parameterization is a powerful way to express all potential solutions in a manageable form. By introducing parameters, we convert qualitative descriptions of solutions into concrete expressions. In our exercise, we let \(x_2 = t\) and \(x_3 = s\), where \(t\) and \(s\) are any real numbers. This parameterization transforms the initial solution expression for \(x_1\) into a format that incorporates \(t\) and \(s\):- \(x_1 = -2t - 3s\)Subsequently, our vector \(\mathbf{x}\) becomes: \[\mathbf{x} = \begin{bmatrix} -2t - 3s \ t \ s \end{bmatrix} = t\begin{bmatrix} -2 \ 1 \ 0 \end{bmatrix} + s\begin{bmatrix} -3 \ 0 \ 1 \end{bmatrix}\] This result shows \(\mathbf{x}\) as a combination of vectors scaled by the parameters \(t\) and \(s\). Each choice of \(t\) and \(s\) represents a unique solution in the nullspace, providing an elegant and thorough description of the infinite solutions available. This demonstrates how parameterization simplifies working with complex solution sets, especially in the context of vector spaces and linear algebra.