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In linear algebra, the nullspace of a matrix is a fundamental concept that helps understand how the matrix transforms vectors. Specifically, the nullspace of a matrix \(A\) is the set of all vectors \(x\) that satisfy the equation \(Ax = 0\). In simpler terms, it's the set of vectors that get mapped to the zero vector by the transformation represented by \(A\). This is critical when analyzing solutions to linear systems of equations, especially when exploring the structure of solutions.To find the nullspace of a matrix, we perform row reduction to bring the matrix to its row-echelon or reduced row-echelon form. With this new form, one can easily identify the free variables that represent the nullspace basis vectors. In the exercise given, after performing row operations on the matrix \(A\), we determine that:

• Choosing \(x_3 = 1\) and \(x_4 = 0\) gives us the vector \([-2, 0, 1, 0]\).
• Choosing \(x_3 = 0\) and \(x_4 = 1\) gives us the vector \([-3, 0, 0, 1]\).

Thus, these two vectors form a basis for the nullspace, meaning every vector in the nullspace can be expressed as a linear combination of these vectors. Understanding the nullspace is vital for analyzing the solutions of homogeneous systems, as it determines the extent of solutions or the number of degrees of freedom in the solution set.

An augmented matrix provides an efficient way to represent and solve a system of linear equations. It combines the coefficients of the variables in the system with the constants from the right-hand side of the equations into a single rectangular array. This allows for easy application of row operations that are used in Gauss-Jordan elimination or Gaussian elimination to simplify the system to a point where solutions can be readily found.The matrix derived from our exercise is represented as:\[\begin{bmatrix}1 & 2 & 0 & 3 & | & b_1 \2 & 4 & 0 & 7 & | & b_2 \\end{bmatrix}\]The vertical bar often symbolizes the boundary between the coefficients of the variables and the constants, representing the equality in the equations.By using row operations, we transform this matrix into a simpler form. The goal is to reach a form that can easily provide the relationship between the variables or, in particular, check the consistency of the system. This was crucial for determining that the system is consistent when \(b_2 = 2b_1\), removing any contradictions.

A system of linear equations consists of multiple linear equations involving the same set of variables. The solutions to these systems represent the values that satisfy all equations simultaneously. Such systems can be solved using numerous methods, including substitution, elimination, and matrix methods like the one applied in the exercise.When represented in matrix form as \(Ax = b\), solving such systems entails finding \(x\), the vector of variables. This process involves transforming the system into a simpler equivalent system by utilizing row operations on the augmented matrix. The ultimate aim is to solve for the variables or determine conditions under which solutions exist.In our exercise, the key task was to determine the conditions on \(b_1\) and \(b_2\) such that a solution exists. By transforming the system's augmented matrix, we found that the system has a solution if and only if \(b_2 = 2b_1\). This condition ensures that there are no inconsistencies when solving the equations. Systems of linear equations are common in many fields, and understanding their properties and solutions is critical for applied mathematics.