Linear Algebra

178 Chapter Three. Maps Between SpacesHowever, if we thereby get an impression that the idea of ‘homomorphism’is in some way secondary to that of ‘isomorphism’ then that is mistaken. In therest of this chapter we shall work mostly with homomorphisms. This is partlybecause any statement made about homomorphisms is automatically true aboutisomorphisms but more because, while the isomorphism concept is more natural,experience shows that the homomorphism concept is more fruitful and morecentral to further progress.Exerciseš 1.17 Decide ⎛ if ⎞each h: R 3 → R 2 is linear. ⎛ ⎞⎛ ⎞x ( )x ( x ((a) h( ⎝xy⎠) =(b) h( ⎝ 0y⎠) = (c) h( ⎝ 1y⎠) =x + y + z0)1)zzz⎛ ⎞x ( )(d) h( ⎝ 2x + yy⎠) =3y − 4zž 1.18 Decide if each map h: M 2×2 → R is linear.( ) a b(a) h( ) = a + dc d( ) a b(b) h( ) = ad − bcc d( ) a b(c) h( ) = 2a + 3b + c − dc d( ) a b(d) h( ) = a 2 + b 2c ď 1.19 Show that these two maps are homomorphisms.(a) d/dx: P 3 → P 2 given by a 0 + a 1 x + a 2 x 2 + a 3 x 3 maps to a 1 + 2a 2 x + 3a 3 x 2(b) ∫ : P 2 → P 3 given by b 0 + b 1 x + b 2 x 2 maps to b 0 x + (b 1 /2)x 2 + (b 2 /3)x 3Are these maps inverse to each other?1.20 Is (perpendicular) projection from R 3 to the xz-plane a homomorphism? Projectionto the yz-plane? To the x-axis? The y-axis? The z-axis? Projection to theorigin?1.21 Show that, while the maps from Example 1.3 preserve linear operations, theyare not isomorphisms.1.22 Is an identity map a linear transformation?̌ 1.23 Stating that a function is ‘linear’ is different than stating that its graph is aline.(a) The function f 1 : R → R given by f 1 (x) = 2x − 1 has a graph that is a line.Show that it is not a linear function.(b) The function f 2 : R 2 → R given by( xy)↦→ x + 2ydoes not have a graph that is a line. Show that it is a linear function.̌ 1.24 Part of the definition of a linear function is that it respects addition. Does alinear function respect subtraction?1.25 Assume that h is a linear transformation of V and that 〈⃗β 1 , . . . , ⃗β n 〉 is a basisof V. Prove each statement.(a) If h(⃗β i ) = ⃗0 for each basis vector then h is the zero map.