## Mathematical Methods. Linear Algebra / Normed Spaces / by Jacob Korevaar

By Jacob Korevaar

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46 1. ALGEBRAIC THEORY OF VECTOR SPACES If <-, -> is any bilinear functional on V we call the function q(x) =

Fn to lFk defined by A. In order to compare the ranks of L and L we COMPARISON OF RANKS. 6. RECTANGULAR MATRICES 59 compare their ranges R and R. Let T denote the isomorphism from W onto 3Fk which takes k k z= Z i i w i into 2 i=l = Σίί^Λ)· i=l The restriction of T to the subspace R of W will still be a one-to-one linear transformation or isomorphism. What is its range ? > ··· ' fc f o r certain £,·, J=l then z is in R, and conversely. Thus T maps 7? onto R. It follows that p(L) = dim R = dim R = p(L) = p(A); the rank of a linear transformation is equal to the rank of any one of its representing matrices.

Prove that the set of all vectors (1, y, y2, -·), y complex, is linearly independent in ^ . What can you say about dim ^ ? 30 1. ALGEBRAIC THEORY OF VECTOR SPACES 4. LINEAR TRANSFORMATIONS A transformation T consists of a domain Z>, a (nameless) set E and a rule which assigns (precisely) one element z = Tx in E to every element x in D (Fig. 4-1). We refer to Tas a transformation from (or of or on) D into (or to) E. The subset of E which consists of the elements z that occur as the image Tx of one or more elements x in Z) is called the range R of the transformation.