System of homogeneous linear equations
Homogeneous linear equations form a specific type of system where all constant terms on the right-hand side of the equations are zero
These equations are of the form:
Characteristics of Homogeneous Equations:
Zero Constant Terms:
- All equations have zero on the right-hand side, resulting in the absence of explicit constant values.
Homogeneity and Linearity:
- The term "homogeneous" refers to the property that the equations are balanced around zero, implying that the solutions form a subspace centered at the origin.
- The equations are linear because they don't contain higher powers of the variables or products between variables.
Properties and Solutions:
- The system always has at least one solution, known as the trivial solution, where all variables equal zero ().
- If a system of homogeneous equations has non-trivial solutions (i.e., solutions where not all variables are zero), it implies that the equations are linearly dependent.
- Linearly dependent equations provide a set of values for variables that satisfy the equations without all variables being zero.
- To determine non-trivial solutions, examine the determinant of the coefficient matrix () and apply row operations to derive the reduced echelon form.
- The number of free variables (variables that can take any value) indicates the dimension of the solution space.
Consider a system of homogeneous equations:
Representing this system in matrix form:
- Determine the number of solutions and whether there are non-trivial solutions based on the determinant and the reduced echelon form.
- Identify the dimension of the solution space and the number of free variables, if any.