# 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
 Author: dsemac91 ID: 184586 Card Set: Test 3 Updated: 2012-11-20 04:38:20 Tags: Linear Algebra Folders: Description: Definitions Show Answers: