Download Linear Algebra Through Geometry (Undergraduate Texts in by John Wermer, Thomas Banchoff PDF
By John Wermer, Thomas Banchoff
This ebook introduces the strategies of linear algebra in the course of the cautious research of 2 and three-d Euclidean geometry. This technique makes it attainable firstly vectors, linear adjustments, and matrices within the context of typical aircraft geometry and to maneuver on to themes equivalent to dot items, determinants, eigenvalues, and quadratic varieties. The later chapters take care of n-dimensional Euclidean area and different finite-dimensional vector house
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Example text
Now look at the general case of a system ax + by = 0, ex + dy = 0, (27) where a, b, e, d are given constants such that not both a and b are zero and not both e and d are zero. Of course, x = 0, y = 0 is a solution of (27). Are there others, and if so what are they? 18 Let x, y be a solution other than 0, O. Set X = (~). Then X*-O and X . (~) = 0 and X· (~) = O. Thus, there is a nonzero vector orthogonal to both vectors (~) and (~). (See Fig. ) This can only happen where (~) and (~) lie on the same line through the origin.
That is, if A, B are linear transformations, then sometimes AB = BA and sometimes AB =1= BA. If AB = BA, we say that A and B commute. For instance, if A is any linear transformation and I is the identity, then A and I commute. Now suppose that A and B are two linear transformations having matrices (~ :) and (: ~), respectively. What is the matrix of the transformation AB? Let X =( ~). Then So AB(X)=A(B (X»= [a c b] [ax bY] _+_ d cx + dy = [a(ax + ~) + b(ex + ~)] c(ax + by) + d(ex + dy) = [(aa +bc)x + (a~ +b~)Y] ( ea + de)x + (cb + dd )y a~ + b~] [x].
Let U be the linear transformation having matrix (a) Interpret U geometrically. (b) Show that UU = I. 3 43 Products of Linear Transformations Exercise 6. Let V be the linear transformation having matrix VV = Ru, rotation by 'IT -6)' Show that (~ radians. Exercise 7. Let Ro and Rq, be rotation by angles of 8 radians and Q radians, "respectively. Show that rotation by 8 + I/> radians. Exercise 8. Exhibit a linear transformation N such that N =F 0, while NN = O. · " . (I2 Exercise 9. Let A be the Imear trans f ormatIOn With matrIX -_ 2I ) .