# Shared Flashcard Set

## Details

MATH415 UIUC
Math415 UIUC
38
Mathematics
Not Applicable
03/13/2019

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 HCombinations of the subspace (u + v) are in HCan 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) REF2) 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 = 02) express solutions x as a linear combination of vectors with the free variables3) 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 = bxp + 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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