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A then r(ab) = (ra)b = a(rb) 0 Thus, an algebra is a vector space in which we can take the product of vectors, or a ring in which we can multiply each element by a scalar (subject, of course, to additional requirements, as given in the definition). We now return to linear transformations. 1 1) The set L(V,W) is a vector space, under ordinary addition of functions and scalar multiplication of functions by elements of F. 2) If u E L(U,V) and T E L(V,W), then the composition ru is in L(U,W). 3) If T E L(V,W) is bijective, then r- 1 E L(W,V).

Formulate and prove a similar statement concerning associativity. Is there an "identity" for direct sum? What about "negatives"? Prove that the vector space 'J of all functions from IR to IR is infinite dimensional. Prove that the vector space e of all continuous functions from IR to IR is infinite dimensional. Let F be a field, and let V be an infinite dimensional vector space over F. What is the cardinality of V? If dim(V) = n does V necessarily contain a subspace of any dimension r satisfying 0 :S r :S n?

S is a maxima/linearly independent set in the sense that S is linearly independent, but any proper superset of S is not linearly independent. Proof. We leave it to the reader to show that {1) and {2) are equivalent. Now suppose (1) holds. Then S is a spanning set. If some proper subset S' of S also spanned V, then any vector in S- S' would be a linear combination of the vectors in S', contradicting the fact that the vectors in S are linearly independent. Hence {1) implies (3). Conversely, if S is a minimal spanning set, then it must be linearly independent.

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A new approach to linear filtering and prediction problems by Kallenrode


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