Math 221: LINEAR ALGEBRA §1-2. Gaussian Elimination Le Chen 1 Emory University, 2020 Fall (last updated on 08/27/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted from those by Karen Seyffarth from University of Calgary.
A matrix is said to be in the row-echelon form (REF) if it a row-echelon matrix. Row-Echelon Matrix ◮ All rows consisting entirely of zeros are at the bottom. ◮ The fjrst nonzero entry in each nonzero row is a 1 (called the leading 1 for that row). ◮ Each leading 1 is to the right of all leading 1 ’s in rows above it. Example 0 1 ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 1 ∗ ∗ ∗ ∗ 0 0 0 0 1 ∗ ∗ ∗ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 where ∗ can be any number.
Row-Echelon Matrix ◮ All rows consisting entirely of zeros are at the bottom. ◮ The fjrst nonzero entry in each nonzero row is a 1 (called the leading 1 for that row). ◮ Each leading 1 is to the right of all leading 1 ’s in rows above it. Example 0 1 ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 1 ∗ ∗ ∗ ∗ 0 0 0 0 1 ∗ ∗ ∗ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 where ∗ can be any number. A matrix is said to be in the row-echelon form (REF) if it a row-echelon matrix.
A matrix is said to be in the reduced row-echelon form (RREF) if it a reduced row-echelon matrix. Reduced Row-Echelon Matrix ◮ Row-echelon matrix. ◮ Each leading 1 is the only nonzero entry in its column. Example 0 1 ∗ 0 0 ∗ ∗ 0 0 0 0 1 0 ∗ ∗ 0 0 0 0 0 1 ∗ ∗ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 where ∗ can be any number.
row-echelon matrix. Reduced Row-Echelon Matrix ◮ Row-echelon matrix. ◮ Each leading 1 is the only nonzero entry in its column. Example 0 1 ∗ 0 0 ∗ ∗ 0 0 0 0 1 0 ∗ ∗ 0 0 0 0 0 1 ∗ ∗ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 where ∗ can be any number. A matrix is said to be in the reduced row-echelon form (RREF) if it a reduced
Examples Which of the following matrices are in the REF? Which ones are in the RREF?
Examples Which of the following matrices are in the REF? Which ones are in the RREF? � 0 � 1 1 0 2 0 � � 1 2 0 0 2 0 (a) (b) (c) 0 0 1 2 0 0 1 2 0 0 1 2 0 0 1 2 � 1 � 1 1 2 0 0 � � 0 2 0 2 0 0 (d) (e) (f) 0 0 1 0 0 1 1 2 0 0 1 2 0 0 0 1
Example Suppose that the following matrix is the augmented matrix of a system of linear equations. We see from this matrix that the system of linear equations has four equations and seven variables. x 1 x 2 x 3 x 4 x 5 x 6 x 7 1 − 3 4 − 2 5 − 7 0 4 0 0 1 8 0 3 − 7 0 0 0 0 1 1 − 1 0 − 1 0 0 0 0 0 0 1 2 Note that the matrix is a row-echelon matrix.
Example Suppose that the following matrix is the augmented matrix of a system of linear equations. We see from this matrix that the system of linear equations has four equations and seven variables. x 1 x 2 x 3 x 4 x 5 x 6 x 7 1 − 3 4 − 2 5 − 7 0 4 0 0 1 8 0 3 − 7 0 0 0 0 1 1 − 1 0 − 1 0 0 0 0 0 0 1 2 Note that the matrix is a row-echelon matrix. ◮ Each column of the matrix corresponds to a variable, and the leading variables are the variables that correspond to columns containing leading ones.
Example Suppose that the following matrix is the augmented matrix of a system of linear equations. We see from this matrix that the system of linear equations has four equations and seven variables. x 1 x 2 x 3 x 4 x 5 x 6 x 7 1 − 3 4 − 2 5 − 7 0 4 0 0 1 8 0 3 − 7 0 0 0 0 1 1 − 1 0 − 1 0 0 0 0 0 0 1 2 Note that the matrix is a row-echelon matrix. ◮ Each column of the matrix corresponds to a variable, and the leading variables are the variables that correspond to columns containing leading ones. ◮ The remaining variables are called non-leading variables.
Example Suppose that the following matrix is the augmented matrix of a system of linear equations. We see from this matrix that the system of linear equations has four equations and seven variables. x 1 x 2 x 3 x 4 x 5 x 6 x 7 1 − 3 4 − 2 5 − 7 0 4 0 0 1 8 0 3 − 7 0 0 0 0 1 1 − 1 0 − 1 0 0 0 0 0 0 1 2 Note that the matrix is a row-echelon matrix. ◮ Each column of the matrix corresponds to a variable, and the leading variables are the variables that correspond to columns containing leading ones. ◮ The remaining variables are called non-leading variables. We will use elementary row operations to transform a matrix to row-echelon (REF) or reduced row-echelon form (RREF).
To solve a system of linear equations proceed as follows: Carry the augmented matrix to a reduced row-echelon matrix using elementary row operations. If a row of the form occurs, the system is inconsistent. Otherwise assign the nonleading variables (if any) parameters and use the equations corresponding to the reduced row-echelon matrix to solve for the leading variables in terms of the parameters. Solving Systems of Linear Equations Theorem Every matrix can be brought to (reduced) row-echelon form by a sequence of elementary row operations.
To solve a system of linear equations proceed as follows: row operations. If a row of the form occurs, the system is inconsistent. Otherwise assign the nonleading variables (if any) parameters and use the equations corresponding to the reduced row-echelon matrix to solve for the leading variables in terms of the parameters. Solving Systems of Linear Equations Theorem Every matrix can be brought to (reduced) row-echelon form by a sequence of elementary row operations. Gaussian Elimination 1. Carry the augmented matrix to a reduced row-echelon matrix using elementary
To solve a system of linear equations proceed as follows: row operations. Otherwise assign the nonleading variables (if any) parameters and use the equations corresponding to the reduced row-echelon matrix to solve for the leading variables in terms of the parameters. Solving Systems of Linear Equations Theorem Every matrix can be brought to (reduced) row-echelon form by a sequence of elementary row operations. Gaussian Elimination 1. Carry the augmented matrix to a reduced row-echelon matrix using elementary 2. If a row of the form [0 0 · · · 0 | 1] occurs, the system is inconsistent.
To solve a system of linear equations proceed as follows: row operations. equations corresponding to the reduced row-echelon matrix to solve for the leading variables in terms of the parameters. Solving Systems of Linear Equations Theorem Every matrix can be brought to (reduced) row-echelon form by a sequence of elementary row operations. Gaussian Elimination 1. Carry the augmented matrix to a reduced row-echelon matrix using elementary 2. If a row of the form [0 0 · · · 0 | 1] occurs, the system is inconsistent. 3. Otherwise assign the nonleading variables (if any) parameters and use the
Gaussian Elimination Problem 2 x + y + 3 z = 1 Solve the system 2 y z + x = 0 − 9 z + x 4 y = 2 −
Gaussian Elimination Problem 2 x + y + 3 z = 1 Solve the system 2 y z + x = 0 − 9 z + x 4 y = 2 − Solution 2 1 3 1 1 2 − 1 0 1 − 4 9 2
Gaussian Elimination Problem 2 x + y + 3 z = 1 Solve the system 2 y z + x = 0 − 9 z + x 4 y = 2 − Solution 2 1 3 1 1 2 − 1 0 → r 1 ↔ r 2 1 2 − 1 0 2 1 3 1 1 − 4 9 2 1 − 4 9 2
Gaussian Elimination Problem 2 x + y + 3 z = 1 Solve the system 2 y z + x = 0 − 9 z + x 4 y = 2 − Solution 2 1 3 1 1 2 − 1 0 → r 1 ↔ r 2 1 2 − 1 0 2 1 3 1 1 − 4 9 2 1 − 4 9 2 1 2 − 1 0 → − 2 r 1 + r 2 , − r 1 + r 3 0 − 3 5 1 0 − 6 10 2
Gaussian Elimination Problem 2 x + y + 3 z = 1 Solve the system 2 y z + x = 0 − 9 z + x 4 y = 2 − Solution 2 1 3 1 1 2 − 1 0 → r 1 ↔ r 2 1 2 − 1 0 2 1 3 1 1 − 4 9 2 1 − 4 9 2 1 2 − 1 0 1 2 − 1 0 → − 2 r 1 + r 2 , − r 1 + r 3 → − 2 r 2 + r 3 0 − 3 5 1 0 − 3 5 1 0 − 6 10 2 0 0 0 0
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