Linear Algebra

2. a set with addition and scalar multiplication on it such that commutativity, associativity, additive identity & inverse, multiplicative identity and distributive properties hold
8. 1v=v ∀v∈V
9. a list of vectors in V that is linearly independent and spans V
10. a vector space is _ if some list of vectors in it spans the space
14. _ of a list (v1,...,vm) of vectors in V is a vector of the form a1v1+...+amvm
15. dimension of the vector space generated by columns of matrix A
16. ∃0∈V such that v+0=v for all v∈V
18. x=(x1,x2) where x is an arrow starting at one point and ending at another
19. a scalar λ∈F of T∈L(V) where there exists a nonzero vector u∈V such that Tu=λu
21. the set of all linear combinations of (v1,...vm)
22. a list (v1,...,vm) of vectors in V is a basis of V iff every v∈V can be written _ in the form v=a1v1+...+amvm) where a1,...,am∈F
23. each vector in V can be uniquely represented in the form u1+...+um where each uj∈Uj for subspaces U1,...,Um of V where V=U1+...+Um
24. <u,v>=0 for vectors u,v∈V

1. u+v=v+u for all u,v∈V
3. ∀v∈V, ∃w∈V such that v+w=0
4. {v∈V|Tv=0}
5. a(u+v)=au+av and (a+b)v=av+bv ∀v,u∈V and a,b∈F
6. for a list (v1,...,vm) of vectors in V the only choice of a1,...,am∈F that makes a1v1+...+amvm equal 0 is a1=...=am=0
7. the subset of V consisting of those vectors that T maps to 0, for T∈L(V,W)
11. (u+v)+w=u+(v+w) and (ab)v=a(bv) for all u,v,w∈V and a,b∈F
12. a subset U of V is a _ of V if U is also a vector space such that 0∈U; u,v∈U implies u+v∈U; a∈F and u∈U implies au∈U
13. a rectangular array with rows and columns
17. u∈V such that Tu=λu for T∈L(V) and λ∈F where λ is and eigenvalue of T
20. the dimension of a finite-dimensional vector space is defined to be the _ of any basis of the vector space
25. a function T:V→W with properties additivity, homogeneity, zero, identity, differentiation, integration, multiplication by x^2, backward shift