By Steven Roman

This graduate point textbook covers a particularly wide variety of issues. The booklet first deals a cautious dialogue of the fundamentals of linear algebra. It then proceeds to a dialogue of modules, emphasizing a comparability with vector areas, and offers a radical dialogue of internal product areas, eigenvalues, eigenvectors, and finite dimensional spectral conception, culminating within the finite dimensional spectral theorem for regular operators. the hot version has been revised and features a bankruptcy at the QR decomposition, singular values and pseudoinverses, and a bankruptcy on convexity, separation and optimistic options to linear systems.

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**Example text**

I Change of Bases for Linear Transformations Since the matrix ·[r]c:B e depends on the ordered bases c:B and e, it is natural to wonder how' to choose these bases in order to make this matrix as simple as possible. For instance, can we always choose the bases so that r is represented by a diagonal matrix? As we will see in Chapter 7, the answer to this question is no.

We will refer to the map 1/Jc:s many times in the sequel. D When two vector spaces are isomorphic, they behave in essentially the same way with respect to the concepts that depend only on linear operations. The following result provides support for this view. 4 Suppose that T E L(V,W) is an isomorphism. Let S be a set of vectors in V, and let r(S) = {r(s) Is E S} be the set of images of the vectors in S. Then 1) S spans V if and only if r(S) spans W. 2) S is linearly independent in V if and only if r(S) is linearly independent in W.

W. 2 Let dim(V) = n. For any ordered basis c:B of V, the map