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MATH415 UIUC
Math415 UIUC
35
Mathematics
Not Applicable
03/13/2019

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Term
Vector Space
Definition
set of vectors which can be added and scaled (without leaving the space!)
Term
Subspace of vector space (H of V)
Definition
The zero vector of V is in H
Combinations of the subspace (u + v) are in H
Can multiply by a scalar and be in H
Term
Nullspace
Definition
Set of solutions Ax = 0
Term
Finding nullspace A
Definition
1) REF augmented matrix with 0 ([A|0)

2) Write solution as linear combination (ex:

(x1,x2,x3) = (1,2,3)x1 + ...

3) Nul(A) = Span (lin comb)
Term
Column space
Definition
Span of the columns
Term
Col(A) is a subspace of the __
Definition
output space R^m
Term
Col(A) is a subspace of the __
Definition
input space (R^n)
Term
If Ax = b, what can you say about b
Definition
b is in Col(A) if a solution exists
Term
Find if span() = R^3
Definition
Augment with a vector and see if consistent
Term
Linear dependence
Definition
If x1v1+x2v2+x3v3 + ... + xpvp = 0, and x1,x2,x3,... are non 0
Term
A single non-zero vectorv1is always __
Definition
linearly independent
Term
Vectors (v1,....,vp) containing the 0 vector are
Definition
linearly dependent
Term
A set of vectors{v1, . . . ,vp} in V is a basis of V if
Definition
V = span(vectors)
vectors linearly independent
Term
V has dimension p if ___
Definition
it has a basis of p vectors
Term
To be a basis of R^n the set must
Definition
Have n elements
Term
Basis for something like:

[image]
Definition
1) REF

2) Find pivot columns.

3) Basis is the columns of ORIGINAL matrix
Term
Find basis for Nul(A)
Definition
1) find the parametric form of the solutions to Ax = 0
2) express solutions x as a linear combination of vectors with the free variables
3) Vectors form the basis

(basically vectors in span are in basis)
Term
Basis for Col(A)
Definition
pivot columns
Term
dim(ColA)
Definition
r (rank=# pivots)
Term
dim(ColA transpose))
Definition
r
Term
dim(NulA)
Definition
n - r
Term
dim Nul A transpose
Definition
m - r
Term
left null space
Definition
null space of A transpose
Term
row space
Definition
column space of A transpose
Term
Inner product
Definition
same as dot product betweeen 2 vectors (ex: v transpose w)
Term
Orthogonal vectors
Definition
v dot w = 0
Term
Orthonormal
Definition
vectors that are unit vectors and orthogonal
Term
v is orthogonal to W if
Definition
it is orthogonal to every vector in W
Term
orthogonal complement
Definition
space of all vectors that are orthogonal to the subspace W
Term
Nul(A) is the orthogonal complement of
Definition
Col A Transpose
Term
Col(A) is the orthogonal complement of
Definition
Nul(A tranpose)
Term
Find all vectors orthognoal to v1 and v2
Definition
Find orthogonal complement to Col(v1 v2), and use nullspace (it is span(nulspace))
Term
Find all vectors orthognoal to v1 and v2
Definition
Find orthogonal complement to Col(v1 v2), and use nullspace
Term
dim(V ) + dim(V^⊥)
Definition
dim(R^n) = n
Term
T(ba) transform
Definition
T(a1)b T(a2)b T(a3)b are the matrix columns
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