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MATH415 UIUC
Math415 UIUC
38
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
Term
Nul(A) and solutions to Ax = b
 Definition
Let Axp = b

xp + Nul(A) will give all solutions in Nullspace.
Term
The columns of A are linearly independent means ___ (3 things)
 Definition
Ax = 0 has only the solution x = 0.
Nul(A) = {0}
A has n pivots
Term
Vectors v1, . . . , vp containing the zero vector are linearly ____
 Definition
dependent
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