All the Mathematics You Missed [college level,bw] by Garrity

By Garrity

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Simply define T*(w*)(v) = w*(T(v)). By the way, note that the direction of the linear transformation T V -7 W is indeed reversed to T* : W* -7 V*. Also by "natural", we do not mean that the map T* is "obvious" but instead that it can be uniquely associated to the original linear transformation T. 10. BOOKS Such a dual map shows up in many different contexts. For example, if X and Yare topological spaces with a continuous map F : X -t Y and if C(X) and C(Y) denote the sets of continuous real-valued functions on X and Y, then here the dual map F* : C(Y) -t C(X) is defined by F*(g)(x) = g(F(x)), where g is a continuous map on Y.

Ami such that T(Vi) = aliWI + ... + amiW m , We want to see that the linear transformation T will correspond to the m x n matrix A = (a~1 aI2 amI Given any vector v in V, with v T(v) = . a mn = aIvI + ... + anvn , we have T(aIVI + +anvn ) + + anT(vn ) al(allWI + + amlW m ) + ... +an(aInWI + + amnw m ). 5. THE DETERMINANT 9 But under the correspondences of the vector spaces with the various column spaces, this can be seen to correspond to the matrix multiplication of A times the column vector corresponding to the vector v: Note that if T : V -+ V is a linear transformation from a vector space to itself, then the corresponding matrix will be n x n, a square matrix.

1 The determinant of a 1 x 1 matrix (a) is the real-valued function det(a) = a. This should not yet seem significant. Before giving the definition of the determinant for a general nxn matrix, we need a little notation. For an n x n matrix denote by Aij the (n - 1) x (n - 1) matrix obtained from A by deleting the ith row and the jth column. For example, if A = (all A '2 = (a21). Similarly if A = 0;D, a21 then A" = (; 12 a ), then a22 n CHAPTER 1. LINEAR ALGEBRA 10 Since we have a definition for the determinant for 1 x 1 matrices, we will now assume by induction that we know the determinant of any (n 1) x (n -1) matrix and use this to find the determinant of an n x n matrix.

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