Ch.1.2- Row Reduction and Echelon forms

What is a "free variable"?

It means the operator is free to choose any value for X3. Each different choice of X3 determines a (different) solution of the system, and every solution of the system is determined by X3.

What are basic variables also known as?

Leading variables, because they correspond to the columns containing leading entries.

What makes it apparent that the solution is not unique?

The fact that free variables are observed in the system of linear equations. e.g. Each different choice of X3 and X4 determine a different solution.

What makes it apparent that there is an existence of a solution?

The fact that the solution of linear systems is consistent, meaning there are no contradictions. e.g. 0 = 4

What are steps 1-4 called of the Row Reduction Algorithm?

The forward phase.

What is a "leading entry of a row"?

The leftmost nonzero entry (in a nonzero row)

What is the definition of "reduced echelon form" or "reduced row echelon form"?

The matrix satisfies the definition of echelon form as well as the following two conditions. 1. The entry in each nonzero row is 1 2. Each leading 1 is the only nonzero entry in its column.

What is an "echelon matrix"?

A matrix that is in echelon form.

What is the DEFINITION 1.2.1. of a "Pivot Position"?

A pivot postion in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is a column of A that contains a pivot position.

What is a basic variable?

A variable that is at is base, meaning it has a value stored within it. e.g. look at the variables X1, X2, and X5.

If a matrix A is row equivalent to an echelon matrix 𝑈, we call 𝑈...?

An echelon form (or row echelon form) of A; If 𝑈 is in reduced echelon form, we call 𝑈 the reduced echelon form of A.

What does "~" mean in linear algebra?

Before a matrix it indicates that the matrix is row equivalent to the preceding matrix.

What are the four steps to the ROW REDUCTION TECHNIQUE?

1. Write the augmented matrix of the system. 2. Use the row reduction algorithm to obtain an equivalent augmented matrix in echelon form. Decides whether the system is consistent. * If there is no solution, stop; otherwise go to the next step. 3. Continue row reduction to obtain the reduced echelon form 4. Write the system of equations corresponding to the matrix obtained in step 3. 5. Rewrite each nonzero equation from Step 4 so that its one basic variable is expressed in terms of any free variables appearing in the equation.

What is partial pivoting?

A computer program selects as a pivot the entry in a column having the largest absolute value. It reduces roundoff errors in the calculations.

What are parametric descriptions of solution sets?

A description of the solution set where the free variables act as parameters.

What was a flop traditionally?

A flop was only a multiplication or division, because addition and subtraction took much less time and could be ignored.

What is the formal definition of THEOREM 2?

A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column-- that is, if and only if an echelon form of the augmented matrix has no row of the form. [ 0 ... 0 b ] with b nonzero If a linear system is consistent, then the solution set contains either ( i ) A unique solution, when there are no free variables, or ( ii ) infinitely many solutions, when there is at least one free variable.

What is the DEFINITION 1.2 of of a rectangular matrix in "Echelon form" or "Row echelon form"?

If the matrix has the following three properties: 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zeros.

What is a pivot?

It is a nonzero number in a pivot position that is used as needed to create zeros via row operations.

What is back substitution?

It is solving the third equation for one of the variables and then plugging it into the second and solving and then plugging that into equation one. Eventually solving all of them.

What is THEOREM 1 of Row Reduction and Echelon forms?

It is the Uniqueness of the Reduced Echelon form: Each matrix row is equivalent to one and only one reduced echelon matrix.

What is a reduced echelon matrix?

One that is in reduced echelon form.

What are the five steps of "The Row Reduction Algorithm"?

Pretend you're solving a rubik's cube. Step 1: Begin with the leftmost nonzero column. This is a pivot column. The pivot position is at the top. Step 2: Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position. Step 3: Use row replacement operation to create zeros in all positions below the pivot. Step 4: Cover (or ignore) the row containing the pivot position and cover all rows, if any, above it. Apply steps 1-3 to the sub matrix that remains. Repeat the process until there are no more nonzero rows to modify. Step 5: Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If a pivot is not 1, make it 1 by a scaling operation.

Any nonzero matrix may be..?

Row reduced, that is transformed by elementary row operations into more than one matrix in echelon form, using different sequences of row operations.

What is the first step in the Row Reduction Algorithm?

Step 1: Begin with the leftmost nonzero column. This is a pivot column. The pivot position is at the top.

What is the second step in the Row Reduction Algorithm?

Step 2: Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position.

What is the third step in the Row Reduction Algorithm?

Step 3: Use row replacement operation to create zeros in all positions below the pivot.

What is the fourth step in the Row Reduction Algorithm?

Step 4: Cover (or ignore) the row containing the pivot position and cover all rows, if any, above it. Apply steps 1-3 to the sub matrix that remains. Repeat the process until there are no more nonzero rows to modify.

What is the fifth step in the Row Reduction Algorithm?

Step 5: Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If a pivot is not 1, make it 1 by a scaling operation.

What is THEOREM 2 known as?

The Existence and Uniqueness theorem.

What procedure outlines how to find and describe all solution of a linear system?

The ROW REDUCTION TECHNIQUE.

When applying step 5 to produce the unique reduced echelon form what is this called?

The backward phase.

What does solving a system amount to?

To finding a parametric description of the solution set or determining that the solution set is empty.

What do we use as parameters for describing a solution set?

Using free variables as the parameters for describing a solution set.

What is a flop?

What an algorithm is measured in (floating point operations). Which is one arithmetic operation (+,-,*,/).