By von Neumann J.
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Additional info for Collected works. Vol.4 Continuous geometry and other topics
2, A(X + Y) = A(X) + A(Y). Thus we see that the operation which takes a number x into the number ax is somehow similar to the operation which takes a vector X into the vector A (X), where A is a linear transformation. Next, consider the equation: ax=y (I) where a and yare given numbers, a "1= 0, and x is an unknown number. We solve (l) by taking the reciprocal 1/ a of a, and multiplying both sides by it, arriving at -d so x -I( ax ) = -I y, an a a = -a1 y. As an analogue of equation (I) for vectors, we may consider a linear transformation A and a vector Y and look for a vector X such that (2) A(X) = Y.
Terms of x, y, where (;:) = (i) P = projection to the line along (;). x +2y x =-5-' I ,_ 2(x y - + 2y) 5 (ii) S is reflection about the line along ,_ 3 x - - SX Y I = . (~). +S 4 y, 4 + 3 SX S y. (iii) Dr is stretching by r. x' = rx, y'=ry. (iv) RIJ is rotation by 0 radians. x' = (cosO)x - (sinO)y, y' = (sinO)x + (cosO)y. 30 Linear Algebra Through Geometry (v) To is the transformation of Example 5. x'= x, y' = X 2 + y. Can we describe some single general type of transformation, expressing x' and y' each in terms of x and y, which includes the above examples as special cases?
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]. ca + de cb + dd Y = [aa + be 42 Linear Algebra Through Geometry So the matrix of AB is (2) ( ali+be a~+b~). cli + de cb + dd We define the product of matrices m(A) and m(B) to be the matrix m(AB) of AB.
Collected works. Vol.4 Continuous geometry and other topics by von Neumann J.