Download Linear Algebra Through Geometry (Undergraduate Texts in by John Wermer, Thomas Banchoff PDF

By John Wermer, Thomas Banchoff

This publication introduces the suggestions of linear algebra during the cautious learn of 2 and third-dimensional Euclidean geometry. This strategy makes it attainable first of all vectors, linear changes, and matrices within the context of regular aircraft geometry and to maneuver on to themes comparable 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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Hence X =(;) =(~). 4 Inverses and Systems of Equations x = 0 is the only vector with A (X) = 0, so (7) holds. The proposition is proved. We saw earlier that if A has an inverse, then (7) holds and so ad - bc =F 0. Let us now proceed in the converse direction. Consider a linear transformation A with matrix ad - bc =F 0. =( f We seek an inverse B for A. Set m(B) (~ ~). n, Assume (8) where p,q,r,s are unknown numbers. We must have (~ ~)(~ ;)=(6 ~), so + br = I, cp + dr = 0, ap (9) and aq cq + bs = 0, + ds = l.

A (E I ) Thus we can de- scribe the matrix of A by saying that its first column is the image of the first basis vector EI and the second column is the image of E 2 • EXAMPLE P(X) 2. Let P denote projection on the line along U = ( ~), so =( ~:~)u. s{}), sm{} then u2 + v 2 sin {} cos (}). 1 allows us to give a simple solution of the following geometric problem: Let A be a linear transformation and let L be a straight line. By the image of L under A we mean the collection of all vectors A (X) when X is a vector whose endpoint lies on L.

Thus ( y'X') = B(X) = So x(u) + (w) = (ux + wy). v y z vx + zy = ux + wy, y' = vx + zy. x' Thus, x', y' have the form of Eq. (1) of this chapter. Hence B is a linear transformation, by definition. The matrix of B is (~ ~). Thus we have proved that if B is a transformation satisfying (6), then B is a linear transformation. 1. Let A be a transformation of the plane. Then A is a linear transformation if and only if for every pair of vectors X and X and every scalar t: A(X + X) = A(X) + A(X), (7a) and A (tX) = tA (X).

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