4- Gaussian Elimination

Question 1

What is a linear equation?
What type of polynomial is it?
What is a linear system?

Answer 1

• a linear combination of variables x1…xn with their respective coefficients a1…an.
• first degree
• a collection of linear equations involving the same variables.

Question 2

What is a solution of the linear system?
What is the solution set of the linear system?
When 2 linear systems are equivalent?

Answer 2

• an ordered list of numbers that make each equation a true statement if these numbers are substituted to x1…xn.
• the set of all possible solutions.
• linear systems with the same solution set.

Question 3

What is the coefficient matrix?
What is the augmented matrix?

Which is the basic strategy to solve s linear system?

Answer 3

• the matrix built with all the coefficients of the variables in the linear system.
• the coefficient matrix with the constant terms.
• replace 1 system with an equivalent system that it is easier to solve.

Question 4

When are 2 matrices row equivalent?

Which are the 3 elementary row operations?

What does it mean that 2 augmented matrices are row equivalent?

Answer 4

• if there is a sequence of elementary row operations that transforms one matrix into the other.

1. Replacement: replace 1 row by the sum of itself and a multiple of another row.
2. Interchange: interchange two rows.
3. Scaling: multiply all entries in a row by a non-zero constant.

• 2 row equivalent augmented matrices have the same solution set, they are equivalent systems.

Question 5

How many solutions can a system have?
When is a system consistent?
When is a system inconsistent?

Answer 5

• 0, 1 or more, infinite
• a system is consistent if it has at least one solution
• a system is inconsistent if it does not have a solution

Question 6

What is a leading entry in a row of a matrix?

When is a rectangular matrix in ROW ECHELON FORM? Which 3 properties does it need to satisfy?

When is it in REDUCED ROW ECHELON FORM? Which 2 more properties does it need to satisfy?

Each mxn matrix is row equivalent to …
- how many echelon form matrices
- how many reduced echelon form matrices?

Answer 6

• it is the left-most non-zero entry.

1. all non-zero rows must be above any rows of all zero.
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 zero.
4. the leading entry in each non zero row is 1.
5. each leading entry is the only non zero entry in its column.

• each matrix is row equivalent to only one reduced echelon form but it is row equivalent to more than one echelon form.

Question 7

What is a pivot position?
What is a pivot column?

Answer 7

• location in the matrix that corresponds to a leading entry in the echelon form of the matrix.
• a column that contains a pivot position.

Question 8

What are the basic variables of an augmented matrix?
What are the free variables of an augmented matrix?

Starting from the reduced echelon form of an augmented matrix, the solution set can be described by…

Answer 8

• variables x1,…,xn corresponding to pivot columns in the matrix.
• variables that are not basic.
• solving the reduced system of equations for the basic variables in terms of the free variables.

Question 9

Looking at the echelon form of a matrix, we can say that the system is inconsistent if…

Answer 9

if the echelon form contains at least one equation of the form 0 = b with b!=0.