Test 3

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  1. 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
  2. Nullity
    The number of free variables associated to a matrix
  3. Rank
    The number of Pivot Columns in a matrix
  4. Basis
    A set of vecots in a Vector space V is a basis if the set is linearly independent and spans V
  5. Vector Space
    Let V be an arbitrary non empty set of objects on which 2 operations are defined: addition and scalar multiplication
  6. 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: Ax = 0}
  7. 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)
  8. 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
  9. Subspace
    • 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)
  10. Dimension
    The dimension of a vector space is the number of vectors in a basis
  11. Equation for Rank 
    • rank = n - nullity
    • same as

    pivots = columns - free vars

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Test 3
2012-11-20 04:38:20
Linear Algebra

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