Linear Algebra

two linear systems are the same if

they have the same solution set

consistent system

either has one solution or infinitely many solutions

inconsistent system

has no solution

rows represent

equations

columns represent

variables

Every elementary row opperation is

Reversible

• never change the solution set of the associated linear system

Echelon form

1. all non-zero rows are above any all zero rows

2. each leading entry of rows is in a column to the right of the leading entry in the row above it

3. all entires in a column below a leading entry are zeros

reduced echelon form

• is unique

• three echelon form properties and:

  4. the leading entry in each nonzero row is 1

  5. each leading 1 is the only nonzero entry in its column

pivot position

the leading entry of a row in a matrix in (reduced) echelon form

pivot column

a column of the matrix that contains a pivot position

free variables indicate that

there are many solutions, not a unique solution

If b is in the span of the columns of A, you can also say that

b is in the column space of A

can a 3×2 matrix span R³?

No, because this matrix is 2D and cannot span a 3D subspace;

• A is unable to have a pivot in every row

can a 2×3 matrix span R²?

yes because it can have a pivot in every row

Homogeneous linear systems

a system of linear equations that can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in R^m 

• always has the trivial solution (x = 0)

if we have a free variable,

we have a non-trivial solution

• thereforem the given vector set is a linearly dependent set

a set of vectors only containing a single vector, v1 is linearly independent if and only if

v1 is not the zero vector

a set containing two vectors {v1,v2} is linearly independent if and only if

neither of the vectors is a multiple of the other

if a set of vectors contains more vectors than their are entries in each vector, then

the set is linearly dependent; automatically gives a free variable

if a vector set in R^n contains the zero vector, then

the set is linearly dependent; nontrivial solution

examples of linear dependence

• more vectors than entries in each row

• set contains the zero vector

[0]

|0|

[0]

identity matrix I is

always nxn (square)

Matrix multiplication:

AB = [Ab1 Ab2 … Abn]

rank(A) is equal to

• the number of nonzero rows in its echelon form B

and/or

• the number of pivot positions (pivot columns) in B

Rank-Nullity theorum

rank(A) + nullity(A) = number of columns of A

the basis for Col(A) is determined by

the pivot columns in the RREF that contain a leading 1

the basis for Nul(A) is determined by

solving for RREFx = 0; just write RREF in parametric vector form

(NulA = the set of all vectors x such that Ax = 0) 

ex: the x for a 3×4 matrix will have four variables (x1, x2, x3, x4)

null space, NulA is

the set of all vectors x such that Ax = 0

If A is invertible then

A^-1 exists and AA^-1 = I (identity matrix)

1 × 1 Identity matrix

[1]

2 × 2 identity matrix

[1 0]

[0 1]

3 × 3 idnetity matrix 

[1 0 0 ]

|0 1 0 |

[0 0 1]