Linear Algebra

Front 1

Linear Equation

Back 1

a1x1 + a2x2 + ... + anxn = b

Front 2

System of Linear Equations

Back 2

Collection of 1 or more linear equations

Front 3

Linear System

Back 3

Collection of 1 or more linear equations

Front 4

Solution

Back 4

A list (s1, s2, ..., sn) of numbers that make an equation a true statement when (s1, s2, ..., sn) are substituted for (x1, x2, ..., xn)

Front 5

Solution Set

Back 5

all possible solutions

Front 6

Two linear systems are equivalent when

Back 6

they share the same solution set

Front 7

Systems of linear equations can have _____ solution(s)

Back 7

no solution
1 solution
∞ solutions

Front 8

Consistent Linear System

Back 8

either:
1 solution
∞ solutions

Front 9

Inconsistent Linear System

Back 9

no solution

Front 10

Augmented matrices of two linear systems are row equivalent. They have the same _____

Back 10

Solution Set

Front 11

Two Fundamental Questions

Back 11

Does at least 1 solution exist?
If a solution exists, is it only one?

Front 12

Unique Solution

Back 12

Only one solution

Front 13

Row Equivalence

Back 13

elementary row operations transforms one matrix into the other

Front 14

Leading Entry

Back 14

the leftmost nonzero entry of a row

Front 15

Matrix in Echelon Form has

Back 15

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

Front 16

Additional Conditions for Reduced Row Echelon Form

Back 16

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

Front 17

Uniqueness of the Reduced Echelon Form

Back 17

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

Front 18

Pivot Position

Back 18

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.

Front 19

Pivot Column

Back 19

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

Front 20

Partial Pivoting

Back 20

Reduction of round-off errors by using largest absolute value when choosing the pivot for a pivot column

Front 21

Basic Variables

Back 21

variables corresponding to pivot columns

Front 22

Free Variable

Back 22

Not Basic Variables

Each different choice of the free variable determines a (different) solution of the system, and every solution of the system is determined by a choice of the free variable.

Front 23

zero vector

Back 23

Bolded 0

Front 24

Weights for:
y = c1v1 + ... + cpvp
are

Back 24

c1, ..., p1

Front 25

The vector field for:
y = c1v1 + ... + cpvp
is

Back 25

y

Front 26

True of False:
The presence of a free variable in a system does not guarantee that the system is consistent.

Back 26

It could be shown the more likely there is a free variable the more likely a system will be inconsistent.

Front 27

Linear Dependence Relation

Back 27

c1v1 + c2v2 + ... + cpvp = 0 is called a linear dependence relation among v1, ... , vp when the weights are not all zero.

Front 28

v1, ..., vp are Linearly Dependent, we mean ...
(pg 56)

Back 28

{v1, ..., vp} is a linearly dependent set

Front 29

v1, ..., vp are Linearly Independent, we mean ...
(pg 56)

Back 29

{v1, ..., vp} is a linearly independent set

Front 30

Linear Dependence

Back 30

Non-Trivial Solution

Front 31

Trivial Solution

Back 31

c1=c2=c3=0

Front 32

Non-Trivial Solution

Back 32

c1=c2=c3≠0

Front 33

Linearly Independent iff

Back 33

Ax = 0 has only the trivial solution