1.1 System of Linear Equations & 1.2

Question 1

What is Linear Algebra?

Answer 1

Linear algebra is the study of vector spaces.

Question 2

System of linear equations(or a linear system)

Answer 2

…is a collection of one or more linear equations involving the same variables - x1 …. x n

Question 3

Solution

Answer 3

A solution of the system is a list (s-1 , …, s-n) of numbers that makes each equation true statement….

Question 4

Solution Set

Answer 4

The set of all possible solutions

Question 5

Equivalent

Answer 5

…

Question 6

Consistent Solution

Answer 6

two possibilities:

(a) exactly one solution
(b) infinite number if solutions

Question 7

Inconsistent

Answer 7

no solution

Question 8

What are the elementary row operations

Answer 8

1. Replacement
2. Interchange
3. Scaling

Question 9

“If the augmented matrices of two linear systems are row equivalent….”

Answer 9

“… then the two systems have the same solution set. “

Question 10

A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties

Answer 10

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 11

A matrix in reduced echelon form or reduced row echelon form:

Answer 11

1-3 above: is a row echelon matrix

4. The leading entry in each nonzero row is 1.
5. Each leading entry 1 is the only nonzero entry in its column.

Question 12

Theorem: Uniqueness of the Reduced Echelon Form

Answer 12

Each matrix is row equivalent to one and only one reduced echelon matrix.

(expand def later)

Question 13

What is a pivot?

Answer 13

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

Question 14

What is a pivot column?

Answer 14

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

Question 15

Describe the Pivot process

Answer 15

Pivot Process:

1. The top of the leftmost nonzero column is the first pivot position. A nonzero entry, or pivot, must be placed in this position.
2. Create zeros below the pivot, 1, by adding multiples of the first row to the rows below, and obtain the next matrix.

Question 16

What does the zero row in a matrix tells you?

Answer 16

The zero row tells you the Linear System will generate an infinite number of solutions.

Question 17

Theorem: Existence and Uniqueness Theorem

Answer 17

A linear system is consistent if and only if an echelon form of its augmented matrix has no row of the form

[ 0 0 0 0 …. 0 b ], where b is not equal to 0.

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.