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Definition
set of vectors which can be added and scaled (without leaving the space!) 


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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 


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1) REF augmented matrix with 0 ([A0)
2) Write solution as linear combination (ex:
(x1,x2,x3) = (1,2,3)x1 + ...
3) Nul(A) = Span (lin comb) 


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Col(A) is a subspace of the __ 

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Col(A) is a subspace of the __ 

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If Ax = b, what can you say about b 

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b is in Col(A) if a solution exists 


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Augment with a vector and see if consistent 


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If x1v1+x2v2+x3v3 + ... + xpvp = 0, and x1,x2,x3,... are non 0 


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A single nonzero vectorv1is always __ 

Definition


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Vectors (v1,....,vp) containing the 0 vector are 

Definition


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A set of vectors{v1, . . . ,vp} in V is a basis of V if 

Definition
V = span(vectors) vectors linearly independent 


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Definition
it has a basis of p vectors 


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To be a basis of R^n the set must 

Definition


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Basis for something like:
[image] 

Definition
1) REF
2) Find pivot columns.
3) Basis is the columns of ORIGINAL matrix 


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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) 


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null space of A transpose 


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column space of A transpose 


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same as dot product betweeen 2 vectors (ex: v transpose w) 


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vectors that are unit vectors and orthogonal 


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it is orthogonal to every vector in W 


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space of all vectors that are orthogonal to the subspace W 


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Nul(A) is the orthogonal complement of 

Definition


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Col(A) is the orthogonal complement of 

Definition


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Find all vectors orthognoal to v1 and v2 

Definition
Find orthogonal complement to Col(v1 v2), and use nullspace (it is span(nulspace)) 


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Find all vectors orthognoal to v1 and v2 

Definition
Find orthogonal complement to Col(v1 v2), and use nullspace 


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T(a1)b T(a2)b T(a3)b are the matrix columns 


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Nul(A) and solutions to Ax = b 

Definition
Let Axp = b
xp + Nul(A) will give all solutions in Nullspace. 


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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 


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Vectors v1, . . . , vp containing the zero vector are linearly ____ 

Definition

