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Column Space
For a matrix A, the set consisting of all the linear combinations of the columns of A is called the column space of A

Nullity
The number of free variables associated to a matrix

Rank
The number of Pivot Columns in a matrix

Basis
A set of vecots in a Vector space V is a basis if the set is linearly independent and spans V

Vector Space
Let V be an arbitrary non empty set of objects on which 2 operations are defined: addition and scalar multiplication

Null Space
For a matrix A, the set of all solutions to the homogeneous equation Ax=0 (zeroVector) that is if A is M x N, then Nul(A)= {x ∈ R^{n }: Ax = 0}

Kernel
The Kernal of a transformation T: V>W is the set of all Vectors in u ∈ V such that T(u) = 0 (zero vector in W)

Range
The Range of a transformation T:V>W is the set of all vectors in W of the form T(x) for some x ∈ V

Subspace
 A Subspace W of a vector space V is a subset of V that satisfies 3 requirements:
 a) The zero vector of v ∈ W
b) if v, w ∈ W then v + w ∈ W
c) if v ∈ W then cv ∈ W, where c is a scalar (closed under scalar multiplication)

Dimension
The dimension of a vector space is the number of vectors in a basis

Equation for Rank
 rank = n  nullity
 same as
pivots = columns  free vars

