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A function for some Te HomF(V, V') f: V -÷ V' is called an affine transformation if f = and some cc cV'. The set of all affine transformations from V to V' will be denoted AffF(V, V'). 10: If Ac d(V) and fe V'), then f(A) e d(V'). E Let us now return to the special subset V/W of d(V). The cosets of W can be given the structure of a vector space. 11, cc and /3 are vectors in V and & and /3 are their corresponding equivalence classes. & -I- if is defined to be the equivalence class that contains cc + /3.

20: F" a)' is a commutative diagram of isomorphisms. 20 is commutative. 15. We need only argue M(b, a) is an -÷ F" is an isomorphism. invertible matrix. Then the map M(b, a)t = SM(La): Now change of basis matrices M(ö, a) are always invertible. 18. For any fJeV, we have M(a, b)M(ö, a)[fl]5 = M(a, b)[fl]a = [fl]5. This equation easily implies M(a, ö)M(ö, a) = I,,, the n x n identity matrix. 6. 21: In this example, we construct a vector space isomorphic to V". Let V be a vector space over F, and let n e N.

Cx(/ly) = for all x, /3, yeA. A2. A4. x(xfl) =(xcx)/3 = x(xfl) for all fleA, xc F. A5. There exists an element I cA such that hx = = for all cxeA. We have seen several examples of (associative) algebras in this book already. and F[X] with the usual Any field F is an associative alebra over F. multiplication of matrices or polynomials is an algebra over F. If V is any vector space over F, then HomF(V, V) becomes an (associative) algebra over F when we define the product of two linear transformations T1 and T2 to be their composite T1T2.

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An Introduction To Linear Algebra by Kuttler

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