Back substitution

Definition

Back substitution is a method used to solve a system of linear equations after it has been transformed into an upper triangular form, typically through Gaussian elimination. This technique involves starting from the last equation and substituting known values back into the preceding equations to find the values of all variables in a sequential manner. It is a crucial step in efficiently solving linear systems, ensuring that all equations are satisfied as values are determined one by one.

5 Must Know Facts For Your Next Test

1. Back substitution starts from the last row of an upper triangular matrix and works upward to find each variable's value.
2. It requires that the system of equations be in a specific form, allowing for straightforward substitution of known values.
3. The process is efficient and can be done quickly, typically in linear time relative to the number of variables.
4. If there are free variables in the system, back substitution may lead to an infinite number of solutions.
5. Understanding back substitution is essential for grasping more complex topics in linear algebra, such as matrix inversion and solving systems with multiple solutions.

Review Questions

• How does back substitution relate to Gaussian elimination, and why is it an important step in solving linear systems?
  • Back substitution is directly related to Gaussian elimination as it is the final step needed to solve the system after transforming it into upper triangular form. Gaussian elimination simplifies the equations, making it easier to isolate each variable. Once in this form, back substitution allows us to find the values of variables starting from the bottom equation and moving upwards, ensuring that we use already known values effectively to satisfy all equations.
• Describe how an upper triangular matrix facilitates the process of back substitution in solving linear equations.
  • An upper triangular matrix simplifies back substitution because all coefficients below the main diagonal are zero. This structure means that when you start solving from the last equation upwards, each equation only contains one unknown variable along with constants and previously solved variables. Consequently, this allows for a clear path to determine each variable sequentially without confusion or backtracking.
• Evaluate the impact of free variables on the outcomes produced through back substitution, particularly in systems with infinitely many solutions.
  • Free variables introduce additional complexity when using back substitution because they can lead to infinitely many solutions in a system of equations. When a variable does not have a leading coefficient in any equation, it can take on any value. As you substitute known values back through the equations, you will find that certain variables can be expressed in terms of these free variables. This creates a situation where multiple combinations of variable values satisfy the original system, illustrating how essential it is to recognize and handle free variables during this process.

Related terms

Gaussian elimination:

A systematic method for solving linear systems by transforming the augmented matrix into an upper triangular form.

Upper triangular matrix: A type of matrix where all entries below the main diagonal are zero, making it easier to perform back substitution.

Row echelon form: A form of a matrix where each leading entry of a row is to the right of the leading entry of the previous row, often a precursor to back substitution.