Linear Algebra Chpt 1

Front 1

Consistent, Inconsistent

Back 1

If at least one solution, no solution

Front 2

Linear system equivalence

Back 2

If they have the same solution sets

Front 3

Row Equivalence

Back 3

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

Front 4

Leading entry

Back 4

refers to the left most non zero entry

Front 5

Pivot postion

Back 5

Is a location in matrix A that corresponds to the leading 1 in a the reduced echelon form of A

Front 6

Thm Existance and uniqueness

Back 6

A linear system is consistent if and only if the right most column of the augment matrix is not a pivot column. if a linear system is consistent then the solution set contains exactly one solution in No Free variables

Front 7

Linear combo of vectors

Back 7

Vector V_1 , V_2...V_n in Rn and C_1, C_2....C_n

V_1C_1+V_2C_2.......

Front 8

Span

Back 8

V1,V2....Vn are vectors in Rn then the set of all linear combinations of V1,....Vn is called the span of V1...Vn and is denoted by {V1....Vn}

Front 9

Homogeneous

Back 9

if it can be written in the form Ax=0 / also has to be consistent

Front 10

Trivial solution

Back 10

Where when Ax=0 and x=0vector

Front 11

Non Trivial

Back 11

When Ax=0 and x does not = 0vector

Front 12

Linear independence

Back 12

A set of vectors Vn in Rn is (independent) if the vector equation c1V1+c2V2....cnVn=0 and there is no non zero c scalar for that equation to be true

Front 13

Linear Dependence

Back 13

A set of vectors Vn in Rn is (independent) if the vector equation c1V1+c2V2....cnVn=0 and there is at least one non zero c scalar that the equation is true.

Front 14

Line Transformation

Back 14

T from Rn to Rm denoted {T: Rn to Rm} is a rule which assigns to each vector X in Rn a vector T(X) in Rm

Front 15

Co-Domain

Back 15

Rn is the domain of T, the transformed set Rm is called the co-domain.

Front 16

Image

Back 16

For vector X in Rn, the vector transformed vector T(X) is called the image of X under T

Front 17

Column Vector

Back 17

Matrix with only one column

Front 18

Vector

Back 18

Matrix with one column

Front 19

Linear combination of vectors A1....An with weights c

Back 19

C1 A1 + C2 A2 .....+CnAn

Front 20

Range in a Transformation

Back 20

The set of all images T(x) is called the range.

Front 21

Linear Transformation

Back 21

a) T(vectorX+vectorY)= T(x) + T(y) for all xy in Rn

b) T(cx)= cT(x) for all x in Rn and scalar C

c) if you get a matrix representation its automatically linear.