Linear Algebra

Section II. Homomorphisms 1832.7 Example We think of R 3 as like R 2 except that vectors have an extracomponent. That is, we think of the vector with components x, y, and z assomehow like the vector with components x and y. In defining the projectionmap π, we make precise which members of the domain we are thinking of asrelated to which members of the codomain.To understanding how the preservation conditions in the definition of homomorphismshow that the domain elements are like the codomain elements,we start by picturing R 2 as the xy-plane inside of R 3 . (Of course, R 2 is notthe xy plane inside of R 3 since the xy plane is a set of three-tall vectors witha third component of zero, but there is a natural correspondence.) Then thepreservation of addition property says that vectors in R 3 act like their shadowsin the plane.⎛ ⎞⎛ ⎞⎛ ⎞x 1 ( ) x 2 ( ) x 1 + y 1 ( )⎝y 1⎠ x1above plus ⎝y 2⎠ x2above equals ⎝y 1 + y 2⎠ x1 + x 2abovey 1 y 2 y 1 + y 2z 1 z 2 z 1 + z 2Thinking of π(⃗v) as the “shadow” of ⃗v in the plane gives this restatement: thesum of the shadows π(⃗v 1 ) + π(⃗v 2 ) equals the shadow of the sum π(⃗v 1 + ⃗v 2 ).Preservation of scalar multiplication is similar.Redrawing by showing the codomain R 2 on the right gives a picture that isuglier but is more faithful to the “bean” sketch.⃗w 2⃗w 1 + ⃗w 2⃗w 1Again, the domain vectors that map to ⃗w 1 lie in a vertical line; the pictureshows one in gray. Call any member of this inverse image π −1 (⃗w 1 ) a “ ⃗w 1 vector.”Similarly, there is a vertical line of “ ⃗w 2 vectors” and a vertical line of “ ⃗w 1 +⃗w 2 vectors.” Now, saying that π is a homomorphism is recognizing that ifπ(⃗v 1 ) = ⃗w 1 and π(⃗v 2 ) = ⃗w 2 then π(⃗v 1 + ⃗v 2 ) = π(⃗v 1 ) + π(⃗v 2 ) = ⃗w 1 + ⃗w 2 . Thatis, the classes add: any ⃗w 1 vector plus any ⃗w 2 vector equals a ⃗w 1 + ⃗w 2 vector.Scalar multiplication is similar.So although R 3 and R 2 are not isomorphic π describes a way in which theyare alike: vectors in R 3 add as do the associated vectors in R 2 — vectors add astheir shadows add.2.8 Example A homomorphism can express an analogy between spaces that is