Linear Algebra

184 Chapter Three. Maps Between Spacesmore subtle than the prior one. For the map from Example 2.6( )x h↦−→ x + yyfix two numbers w 1 , w 2 in the range R. A ⃗v 1 that maps to w 1 has componentsthat add to w 1 , so the inverse image h −1 (w 1 ) is the set of vectors with endpointon the diagonal line x + y = w 1 . Think of these as “w 1 vectors.” Similarly wehave “w 2 vectors” and “w 1 + w 2 vectors.” The addition preservation propertysays this.⃗v 1 ⃗v2⃗v 1 + ⃗v 2a “w 1 vector” plus a “w 2 vector” equals a “w 1 + w 2 vector”Restated, if we add a w 1 vector to a w 2 vector then h maps the result to aw 1 + w 2 vector. Briefly, the sum of the images is the image of the sum. Evenmore briefly, h(⃗v 1 ) + h(⃗v 2 ) = h(⃗v 1 + ⃗v 2 ).2.9 Example The inverse images can be structures other than lines. For the linearmap h: R 3 → R 2⎛ ⎞x ( )⎜ ⎟ x⎝y⎠ ↦→xzthe inverse image sets are planes x = 0, x = 1, etc., perpendicular to the x-axis.We won’t describe how every homomorphism that we will use is an analogybecause the formal sense that we make of “alike in that . . . ” is ‘a homomorphismexists such that . . . ’. Nonetheless, the idea that a homomorphism between twospaces expresses how the domain’s vectors fall into classes that act like therange’s vectors is a good way to view homomorphisms.Another reason that we won’t treat all of the homomorphisms that we see asabove is that many vector spaces are hard to draw, e.g., a space of polynomials.But there is nothing wrong with leveraging those spaces that we can draw. Wederive two insights from the three examples 2.7, 2.8, and 2.9.The first insight is that in all three examples the inverse image of the range’szero vector is a line or plane through the origin, a subspace of the domain.