Dual Spaces. Let V and W be finite dimensional vector spaces over a field F. Sho

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Dual Spaces. Let V and W be finite dimensional vector spaces over a field F. Show that the set of all linear transformations from V into W , denoted by Hom(V, W ), is a vector space over F ,where we define vector addition as follows: (S +T)(v) =S(v) +T(v) ( alpha S)(v) = alpha S(v), where S,T Hom(V, W) alpha F, and v V . Let V be an F -vector space. Define the dual space of V to be V * = Hom(V, F ). Elements in the dual space of V are called linear functionals. Let vl,..., vn be an ordered basis for V. if v = alpha 1 v1 +...+ alpha n Vn is any vector in V , define a linear functional phi: V rightarrow F by phi (v) = alpha i. Show that the (pi's form a basis for V * . This basis is called the dual basis of vl,..., vn (or simply the dual basis if the context makes the meaning clear). Consider the basis {(3,1), (2, -2)} for R2 . What is the dual basis for (R2 )* ? Let V be a vector space of dimension n over a field F and let V * * be the dual space V * . Show that each element v V gives rise to an element lambda v in V ** and that the map v rightarrow lambda v is an isomorphism of V with V **Direct Sums

Explanation / Answer

Let the tranformation be T

T(v) --> w

nowAvector spaceor linear spaceV, is a set which satisfies the givne conditon for allu,vandwinVand scalars c and d

b)

Anyvector space,V, has a correspondingdual vector space(or justdual spacefor short) consisting of alllinear functionsonV. Dual vector spaces defined on finite-dimensional vector spaces can be used for definingtensors

I can define complete vector space for you but for that u need to rate first since I have lost many points even after complete answering..

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