Linear Algebra Through Geometry (Undergraduate Texts in Mathematics) (Hardcover)

by Thomas Banchoff (Author), John Wermer (Author) "Analytic geometry begins with the line..." (more)
Key Phrases: oriented triplet, linear transformation with matrix, first coordinate axis, Linear Algebra Through Geometry, Spectral Theorem, The Matrix Relative (more...)

Editorial Reviews

Product Description
Linear Algebra Through Geometry introduces the concepts of linear algebra through the careful study of two and three-dimensional Euclidean geometry. This approach makes it possible to start with vectors, linear transformations, and matrices in the context of familiar plane geometry and to move directly to topics such as dot products, determinants, eigenvalues, and quadratic forms. The later chapters deal with n-dimensional Euclidean space and other finite-dimensional vector space. Topics include systems of linear equations in n variable, inner products, symmetric matrices, and quadratic forms. The final chapter treats application of linear algebra to differential systems, least square approximations and curvature of surfaces in three spaces. The only prerequisite for reading this book (with the exception of one section on systems of differential equations) are high school geometry, algebra, and introductory trigonometry.

Product Details

• Hardcover: 305 pages
• Publisher: Springer; 2nd edition (September 10, 1993)
• Language: English
• ISBN-10: 0387975861
• ISBN-13: 978-0387975863
• Product Dimensions: 9.2 x 6.2 x 1 inches
• Shipping Weight: 1.3 pounds (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars See all reviews (2 customer reviews)
• Amazon.com Sales Rank: #503,878 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

12 of 12 people found the following review helpful:

5.0 out of 5 stars A wonderful introduction, July 21, 2005

By	Nicholas Warren (New York, NY USA) - See all my reviews (REAL NAME)

A crystal clear book. It shows the geometrical significance of topics like eigenvalues/eigenvectors without losing the intuition in the formalism, like so many other books on Linear Algebra. The book would be particularly suitable for scientists, though prospective mathematicians would appreciate the "whys" too.

I only wish there were follow-on, more advanced, volumes!

5 of 8 people found the following review helpful:

3.0 out of 5 stars Unreadable, October 6, 2007

By	Viktor Blasjo - See all my reviews (REAL NAME)

This is a very poorly written and highly opaque book. For example, chapter 2.7 (sic) is called "Classification of Conic Sections", but conic sections are not even mentioned until five pages into this 12-page chapter. Apparently, Banchoff and Wermer finds it appropriate that a reader who wants to learn about the classification of conic sections should be forced to wade through five pages of technicalities, including things like six exercises on computing powers of various matrices, without the slightest indication of what this has to do with the classification of conic sections. Another instance, one among many, where Banchoff and Wermer demonstrate their commitment to technical nonsense and aversion to broad understanding is chapter 2.5 on determinants. "The quantity ad-bc is called the determinant", etc., and then the opening paragraph ends "We shall see that the determinant gives us further information about the behaviour of A" (p. 61), and there follows a devastatingly boring and longwinded five-page discussion on the orientation of an ordered pair of vectors, which is a technicality that could and should be dismissed in one paragraph. Only after this do we see that determinants have to do with area, which is the defining geometric property of the determinant, and of enormous importance. Why not do it the other way around? Why not say straight away that determinants are areas and then deal with the technical matter of orientation and signed areas at the end of the section, instead of keeping the reader in the dark with the secretive and mysterious proclamation that the determinant "gives us further information about the behaviour of A"? Banchoff and Wermer also adhere to a most unfortunate dichotomy between two, three, and n dimensions, which makes the book clumsily structured and repetitive. So, for example, there are three sections called "Linear transformations and matrices" (2.2, 3.2, 4.2), which all do basically the same thing, but our authors still pretend in 4.2 that we know nothing about linear transformations and matrices: "The symbol [(a_ij)] is called the matrix of the transformation ... Any transformation which can be written in this form is called a linear transformation", etc. (p. 213).