Test 3

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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 ∈ Rⁿ: 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