Busy. Please wait.
or

Forgot Password?

Don't have an account?  Sign up
or

taken

why

Make sure to remember your password. If you forget it there is no way for StudyStack to send you a reset link. You would need to create a new account.

By signing up, I agree to StudyStack's Terms of Service and Privacy Policy.

Already a StudyStack user? Log In

Reset Password
Enter the associated with your account, and we'll email you a link to reset your password.

Remove ads
Don't know
Know
remaining cards
Save
0:01

 Flashcards Matching Hangman Crossword Type In Quiz Test StudyStack Study Table Bug Match Hungry Bug Unscramble Chopped Targets

Embed Code - If you would like this activity on your web page, copy the script below and paste it into your web page.

Normal Size     Small Size show me how

AA2 Chap 19

QuestionAnswer
Vector Space A vector space V over a field F, is an Abelian group under addition and closed over scalar multiplication. The elements of F are called SCALARS The elements of V are called VECTORS
Examples of Vector Spaces R^n is a vector space over R M(2x2) is a vector space over R C is a vector space over R
Basis for V A BASIS for V is a set of vectors that are linearly independent and span V.
linearly independent v1,v2,...,vn are linearly independent if the only solution to c1v1+c2v2+...+cnvn=0, is ci=0
span v1,v2,...,vn SPAN V if for all v in V there exists ci such that c1v1+c2v2+...+cnvn=v
A basis is a... minimal spanning set
A basis is a..... maximal linearly independent set
Theorem about Vector spaces and Bases Every vector space has a basis Proof: Zorn's Lemma
Vector Space letters For all v in V, a in F, av in V, 1) a(v+w)=av+aw 2)(a+b)v=av+bv 3)(ab)v=a(bv) 4)1*v=v
Subspace U is a SUBSPACE of V if U is a subset that is also a vector space over F.
Subspace spanned Let v1,v2,...,vn be in V, then <v1,v2,...,vn>={a1v1+a2v2+...+anvn|ai is in F} is called a subspace spanned by v1,v2,...vn.
Linear combination a1v1+a2v2+...+anvn is called a linear combination of v1,v2,...,vn if V=<v1,v2,...,vn>, then {v1,v2,...vn} spans V. {v1,v2,...,vn} is a spanning set for V.
Linearly independent v1,v2,...,vn are LINEARLY INDEPENDENT over F if there exists a1,a2,...,an in F (not all 0) such that a1v1+a2v2+...+anvn=0
Created by: csebald