Systems of Linear Equations and Gaussian Elimination

Question 1

What is a system of linear equations?

Answer 1

A system of linear equations is a collection of one or more linear equations involving the same variables.

Question 2

The set of all possible solutions of a linear system is called …

Answer 2

… the solution set of the linear system.

Question 3

When are two linear systems equivalent?

Answer 3

When they have the same solution set.

Question 4

There are three possibilities when it comes to arriving at a solution for a system of linear equations.
What are thoooooose?

Answer 4

1. no solution (graphically, when the function lines are parallel, they never intersect)
2. exactly one solution (the lines intersect in one point)
3. infinitely many solutions (the lines overlay at some point)

Question 5

What is the difference between a coefficient matrix and a augmented matrix?

Answer 5

The coefficient matrix only includes the values from the system of linear equation on the left side of the equal signs, that is, only the ones corresponding to the x’s.

The augmented matrix consists of the coefficient matrix with an added column containing the constants from the right side of the equal signs.

Question 6

“3 x 4” matrix. Read it and say how many columns and rows.

Answer 6

“three by four matrix”

three ROWS
four COLUMNS

Question 7

What is the basic strategy to solve a system of linear equations? (You do that through iterations every time)

Answer 7

The basic strategy is to replace the system with an equivalent that is easier to solve.

Question 8

What are the three basic operations that are used to simplify a linear system?

Answer 8

1. Replace one equation by the sum of itself and a multiple of another equation (REPLACEMENT)
2. Interchange two equations (INTERCHANGE)
3. Multiply all terms in an equation with a non-zero constant (SCALING)

Question 9

When are two matrices called row equivalent?

Answer 9

Two matrices are called row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other.

Question 10

When is a system of linear equations consistent?

Answer 10

A consistent system of equations has at least one solution, and an inconsistent system has no solution.

Question 11

In a nonzero row (having at least one entry that is not 0) what is a leading entry?

Answer 11

A leading entry of a row refers to the leftmost nonzero entry.

Question 12

A rectangular matrix is in echelon form if it has the following three properties:
…

Answer 12

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

Question 13

Additional to the three properties that a matrix has to fulfil to be in echelon form, what other two properties does it need to fulfil to be in reduced echelon form?

Answer 13

4. The leading entry in each nonzero row is 1

5. Each leading entry is the only nonzero entry in its column

Question 14

What is an echelon matrix?

Answer 14

A matrix that is in echelon form.

Question 15

What is a diagonal matrix?

Answer 15

A n x n matrix whose non-diagonal entries are 0.

Question 16

Each matrix is row equivalent to one and only one reduced echelon matrix. True or false?

Answer 16

True.

Any nonzero matrix may be row reduced into more than one matrix in echelon form. However, the reduced echelon form one is UNIQUE.

Question 17

What is a pivot position?

Answer 17

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

Question 18

What is a pivot column?

Answer 18

A pivot column is a column of a matrix that contains a pivot position.

Question 19

There is an easy way to deduce which are basic variables and which are free when you have reduced a matrix to echelon form. How?

Answer 19

Basic variables correspond to the pivot columns in the matrix.

The ones that do not are called free variables.

Question 20

What does “free variable” mean? And, what does it imply?

Answer 20

When a variable is free, you are free to choose any value for the variable.

Each different choice of the variable value determines a different solution of the system.

Question 21

What does the existence of a free variable tell us?

Answer 21

That the solution of the system of linear equation is not unique.

Question 22

If the rightmost column of the augmented matrix is a pivot column, or if there exists a line in the echelon form matrix that has only 0 0 0… b, then …

Answer 22

The system is inconsistent.

Question 23

A linear system is consistent when the solution set contains either

(i) ________ solution/s, when there are no ___________ variables; or
(ii) ____________ ___________ solutions, when there is ____________ one free variable

Answer 23

A linear system is consistent when the solution set contains either

(i) a unique solution/s, when there are no __free__ variables; or
(ii) __infinitely__ __many___ solutions, when there is ___at least___ one free variable