key term - Echelon Form

Definition

Echelon form is a special matrix form that arises in the process of Gaussian elimination, a method used to solve systems of linear equations. It represents the transformed matrix where the leading entry in each row is 1, and all other entries in that column are 0, creating a stair-step pattern.

5 Must Know Facts For Your Next Test

1. The echelon form of a matrix is obtained by performing a sequence of elementary row operations on the original matrix.
2. Echelon form is useful for determining the rank of a matrix, which is the number of linearly independent rows or columns.
3. The echelon form of a matrix can be used to find the solutions to a system of linear equations, as the variables corresponding to the free columns can be expressed in terms of the other variables.
4. Echelon form is a unique representation of a matrix, meaning that two matrices have the same echelon form if and only if they represent the same system of linear equations.
5. The reduced row echelon form of a matrix is a further transformation of the echelon form, where the leading entry in each row is 1, and all other entries in that column are 0.

Review Questions

• Explain the purpose of Gaussian elimination and how it relates to the echelon form of a matrix.
  • Gaussian elimination is a method used to solve systems of linear equations by performing row operations on the coefficient matrix to transform it into row echelon form. The echelon form of the matrix represents the transformed matrix where the leading entry in each row is 1, and all other entries in that column are 0, creating a stair-step pattern. This transformation allows for the efficient identification of the solutions to the system of linear equations, as the variables corresponding to the free columns can be expressed in terms of the other variables.
• Describe how the echelon form of a matrix can be used to determine the rank of the matrix.
  • The rank of a matrix is the number of linearly independent rows or columns in the matrix. The echelon form of a matrix can be used to determine its rank because the number of non-zero rows in the echelon form is equal to the rank of the matrix. This is because the row operations performed during Gaussian elimination to obtain the echelon form do not change the rank of the matrix. By examining the echelon form, you can easily count the number of non-zero rows, which directly corresponds to the rank of the original matrix.
• Explain the significance of the reduced row echelon form of a matrix and how it differs from the echelon form.
  • The reduced row echelon form of a matrix is a further transformation of the echelon form, where the leading entry in each row is 1, and all other entries in that column are 0. This form is significant because it provides a unique representation of the matrix, meaning that two matrices have the same reduced row echelon form if and only if they represent the same system of linear equations. The reduced row echelon form is more refined than the echelon form, as it ensures that the leading 1 in each row is the only non-zero entry in its column, whereas the echelon form may still have non-zero entries in the columns of the leading 1s.

Related terms

Gaussian Elimination:

A method for solving systems of linear equations by performing row operations to transform the coefficient matrix into row echelon form.

Row Echelon Form:

A matrix form where the leading entry in each row is 1, and all other entries in that column are 0, creating a stair-step pattern.

Augmented Matrix:

A matrix formed by appending the constant terms (right-hand side) of a system of linear equations to the coefficient matrix.