Term
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Definition
a vector v in a vector space V of the vectors u1, u2,...uk in V when v can be written in the form
v=c1u1+c2v2+...ckvk
where c1,c2,...ck are scalars |
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Term
example of a linear combination system:
determine if w=(1,-2,2) is a linear combination of vectors in S={(1,2,3),(0,1,2),(-1,0,1) |
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Definition
(1,-2,2)=a(1,2,3)+b(0,1,2)+c(-1,0,1)
(use G-J elimination) |
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Term
spanning set of S
(let S={v1,v2...vk) be a subset of vector space V) |
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Definition
| when every vector in v can be written as a linear combination of the vectors of S. in such cases it is said that S spans V. |
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Term
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Definition
the set of all linear combinations of the vectors in S (if S={v1,v2,...vk} is a set of vectors in a vector space)
i.e. span(S)={c1v1+c2v2+,...ckvk} , c's are all real numbers |
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Definition
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Term
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Definition
| span{v1,v2,...vk} or span(s) |
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Term
span(S) is a subspace of V
(theorem 4.7) |
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Definition
| if S={v1,v2,...vk} is a set of vectors in a vector space V, then span(S) is a subspace of V. (every other subspace of V that contains S must contain span(S)) |
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Term
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Definition
when a set of vectors S={v1,v2,...vk} in vector space V's vector equation
c1v1+c2v2+...ckvk=0(vector)
has only the trivial solution
c1=0, c2=0, c3=0 |
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Definition
| when there are nontrivial solutions of S |
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Term
| testing for linear (In)dependence |
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Definition
1) look at the vector equation 0=c1v1+c2v2+...+cnvn. write a system of linear equations in with variables c1, c2, cn
2) use Gaussian elimination to determine whether the system has a unique solution
3) if the system has only the trivial solution, then the set is LI. if the system also has a nontrivial solution then the set is LD |
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Term
| property of linearly dependent sets |
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Definition
| a set S={v1,v2,...vk} (k≥2) is lindearly dependent if and only if at least one of the vectos vj can be written as a linear combination of the other vectors in S |
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Term
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Definition
| two vectors u and v in a vector space v are linearly dependent if and only if one is a scalar multiple of the other |
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