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#flo #hw #reading
1 Bases !
- types of lists, so far
- linearly independent lists
- spanning lists
- list of vecs, that when span()-ed, contains all the elements of the vector space.
- ie. you can use linear combo of the original list to get to every element in the vector space
title: basis a *basis* of V is a list of vectors in V that is [linearly independant](KBxLinearIndependence) and [spans](KBxSpansLinAlg) V
such as,
standard basis of $F^n$ is $(1,0, ... , 0), (0,1, ... , 0) , ... , (0, ... , 0,1)$ $(1,2),(3,5)$ -> basis of $F^2$
- things can have many basis!
1.0.1 criterion for basis
title: criterian for basis a list $v_1, ... , v_n$ of vectors in V is a basis of V iff every $v \in V$ can be written uniqely in the form $$ v = a_1 v_1 + ... + a_n v_n$$ where $a_1, ... , a_n \in F$
essentially, for a list of vectors in V to be a basis of V, every element in V has to be written uniqely as the linear combo of the org list of vectors. uh, #review
1.0.2 spanning lists and basis
- spanning list isnt nesasarrily a basis cus they don't need to be
linearly independent
- but, each spanning list does contain a basis
- each spanning list can be converted to a basis through the removal of some number of elements
- but also, every linearly independent list extends to a basis
- can be extended to a basis