Linear Algebra

A is an invertible, NxN matrix

For all B in R^N, Ax=b has the unique solution x=(A^-1)b

A has N pivot columns

The Columns of A Span R^N

N Pivot Positions

b is a linear combination of the columns of A

x|->Ax maps R^N ONTO R^N

At least ONE solution

x|->AX is ONE-TO-ONE

At Most One Solution

Ax=0 has only the trivial solution

Linearly Independent

The columns of A do not contain the Zero Vector

The Columns of A are Linearly Independent

A^T is invertible

A is row equivalent to I sub n

Let C and D be NxN Matrices

CA=I; AD=I

C=D

Let A be an augmented matrix

If the rightmost column of A is a pivot column

There must exist a row [0 0 ... 0 b]

The System is inconsistent

b is not a linear combination of the columns of A

The System has no solutions

b is not a linear combination of the columns of A

If the rightmost column is not a pivot column

Check for Free Variables

If A contains one or more free variables

The System has Infinitely Many Solutions

Ax=0 has a nontrivial solution

Writing the Solution Set of a Consistent System in Parametric Vector Form

1. Row Reduce the Augmented Matrix to Reduced Eschelon Form

2. Express each basic variable in terms of any free variables appearing in an equation

3. Write a typical solution x as a vector whose entries depend on the free variables, if any

4. Decompose x into a linear combination of vectors (with numeric entries) using the free variables as parameters.

If A contains ONLY basic variables

The System has a Unique Solution

Ax=0 has the trivial solution

Let A be size MxN: The following are logically equivalvent

For each b in R^M, the equation Ax=b has a solution

Each b in R^M is a linear combination of the columns of A

The columns of A span R^M

A has a pivot position in every row

Linear Independence

X1V1+X2V2+...+XpVp=0 has only the trivial solution

Ax=0 has only the trivial solution

Linear Dependence

There exist weights, C1...Cp, not all zero, such that C1V1+C2V2+...+CpVp=0

One or Two Vectors

One vector is the scalar multiple of another in a set of two vectors

Two or More Vectors

One vector in an arbitrary set S is a linear combination of the others

A set contains more vectors than entries in each vector. That is, any set {V1,...,Vp} in R^N is linearly dependent if P>N

A set S={V1,...,Vp} in R^N contains the zero vector

A=[a b, c d]

If ad-bc is not 0, then A is invertible

If A is an invertible NxN matrix, then for each b in R^N, the equation Ax=b has the unique solution x=(A^-1)b

If ad-bc=0, then A is NOT invertible

A^-1=(1/ad-bc) [d -b,-c a]

Determinant: ad-bc=det A

A Transformation, T, is linear if

T(u+v)=T(u)+T(v) for all u,v in the domain of T

T(cu)=cT(u) for all u and scalars c

Note: for all vectors u,v in the domain of T and all scalars c,d

T(0)=0

T(cu+dv)=cT(u)+dT(v)

Standard Matrix of a Linear Transformation

T(x)=Ax for all x in R^N

A= [T(e1) T(e2) ... T(en)]

ONTO

each b in R^M is the image of at least one x in R^N

Ax=b has at least one solution

The Columns of A span R^M

ONE-TO-ONE

each b in R^M is the image of at most one x in R^N

Ax=b has at most one solution

T(x)=0 has only the trivial solution

The columns of A are linearly independent

Ly=b

Ux=y