Systems of Linear Equations and Gaussian Elimination

Question 1

What kind of solutions can a system of linear equations have?

Answer 1

1. no solution
2. one solution
3. infinitely many solutions

Question 2

How can you solve linear systems?
i.e. x1 - 2x2 + x3 = 0
2x2 - 8x3 = 8

Answer 2

using a matrix also called coefficient matrix
1 -2 1 0
0 2 -8 8

Question 3

What possible actions can you take to solve a linear system?

Answer 3

1. replace on equation by a sum of itself and a multiple of the other equation
2. inter change two rows
3. multiply all terms in an equation with a non-zero constant

Question 4

What are the two fundamental questions about a linear system?

Answer 4

1. Is the system consistent that is does at least one solution exist
2. If a solution exists is it the only one that is is the solution unique?

Question 5

What three properties have to be fulfilled for a matrix to be in row echelon form?

Answer 5

1. All non-zero rows are above any rows of all zeros
2. Each leading entry of a row is one 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

Question 6

When is a matrix in reduced row echelon form?

Answer 6

when it fulfills all conditions for the row echelon form and additionally

• the leading entry in each nonzero row is 1
• each leading 1 is the only non-zero entry in its column

Question 7

What is a pivot position in a matrix A?

what is a pivot column?

Answer 7

the position of a leading one in the reduced row echelon form
A pivot column is a column of A that contains a pivot position

Question 8

does a 0 = 0 line in a linear system make it inconsistent?

Answer 8

NO!

Question 9

What are basic variables?

Answer 9

the variables corresponding to pivot columns are called basic variables

Question 10

what are free variables?

Answer 10

The variables that are non-basic variables (do not correspond to pivot columns)