By Ciro Ciliberto
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Extra info for Algebra Lineare
4 Inverses and Systems of Equations § 1 Inverses If a, x, yare numbers, then a(x + y) = ax + ay. 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.
_) (x+x) y+y = (ae db)(X+X) y+y =(a(x + x) + bey + y») = ((ax + by) + (ax + bY») e(x + x) + dey + y) (ex + dy) + (eX + dJi) b)(X) d Y = A(X) + A (X). A X+X =A Thus, we have found A(X + X) = A (X) + A (X) (4) for every pair of vectors X, X. A similar calculation shows A (IX) = IA (X), if X is a vector and I is a scalar. Exercise 1. Verify that formula (5) is true. 2 33 Linear Transformations and Matrices Conversely, let B be a transformation of the plane. , suppose + X) = B(X) + B(X), B(tX) = tB(X), (6) whenever X and X are vectors and t is a scalar.
By definition (2), and so ( pr q)(X)_(px+qy) s y - rx + sy . Formula (3) is basic. We interpret (3) as saying that the matrix ( on the vector (;) to yield the vector ( ~: : if). f n (3) acts 32 Linear Algebra Through Geometry EXAMPLE 1. ), G~)( ~) = (2X ~ 2Y)' (~ ~)(~) = (~). Let A be an arbitrary linear transformation. , A (0) = 0, for if (~ ~) is the matrix of A, then A(O)=(~ ~)(~)=(~)=O. A basic reason why linear transformations are interesting is that a linear transformation acts in a simple way on the sum of two vectors.
Algebra Lineare by Ciro Ciliberto