Test 3

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

The number of free variables associated to a matrix

The number of Pivot Columns in a matrix

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

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

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}

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)

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

• 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)

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

• rank = n - nullity
• same as

pivots = columns - free vars