Test 3
Card Set Information
Author:
dsemac91
ID:
184586
Filename:
Test 3
Updated:
2012-11-19 23:38:20
Tags:
Linear Algebra
Folders:
Description:
Definitions
Show Answers:
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