Flashcards
vector spaces
Cards in this deck (11)
- 1vector spaceslet V be an arbitrary non empty set of objects on which 2 operations are defined, addition and scalar multiplication
- 2subspacesa subset W of a vector space is a subspace of V iff
a)if u & v are vectors in W u+v is in W
b)if k is ANY scalar and u is in W then ku is in W - 3TRIVIAL SUBSPACESzero vector and W is a trivial subspace of W
- 4IS IT A SUBSPACE..........IS IT REALLY?????1)luk for values tt satisfy the given equation and prove that the space is non empty
2)give arbitrary vectors
3)and then substitute in original equation
- 5linear combinationsw is a linear combo if =k1v1 +k2v2..........+krvr
matrix of coefficients must be constistant
(that means that it must have a non -zero determinant) - 6spanif v1,v2.....,vr are vectors in vector space V then the set W of all linear combinations of these is a subspace of V and spans V
(THE COMBINATIONS SPAN THE ORIGINAL
NB the v1 ,v2 can also span) - 7to cheq if a set of vectors span VCHEQ FOR CONSISTENCY
non 0 determinent meaning t actually has slns coz we wld hev equated coefficient matrix to an arbitrary vector b - 8linear independencereduce to row echelon form(patrickjmt)
then determine if non trivial slns exist
eg v1=2k, v2=k, v3=1/2k - 9BASIS (in a general sense)vectors that form the coordinate system for space
if S is a basis for a vector space then t can be expressed in exactly 1 way{LI}
and it spans V - 10BASIS (from a mathematical view)to show that gvn vectors are a basis
FIND DETERMINANT(non 0 for consistancy)
proves both LI and spanning
to find basis
GAUSS-JORDAN
formulate eqns en subs them in coefficient matrix only to separate them to find mabasis - 11basis for row space and basis for column spacefor row space is the one we wr looking for in prev card
bt for column space the tt the column corresponding to the one with leading 1s in gauss-jordaned matrix
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- Engineering Flashcards on vector spaces, created by shamie mugaradziko on 16/03/2018.
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- shamie mugaradziko
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