Linear Algebra

Section II. Homomorphisms 179(b) If h(⃗β i ) = ⃗β i for each basis vector then h is the identity map.(c) If there is a scalar r such that h(⃗β i ) = r·⃗β i for each basis vector then h(⃗v) = r·⃗vfor all vectors in V.̌ 1.26 Consider the vector space R + where vector addition and scalar multiplicationare not the ones inherited from R but rather are these: a + b is the product ofa and b, and r · a is the r-th power of a. (This was shown to be a vector spacein an earlier exercise.) Verify that the natural logarithm map ln: R + → R is ahomomorphism between these two spaces. Is it an isomorphism?̌ 1.27 Consider this transformation of R 2 .( xy)↦→( ) x/2y/3Find the image under this map of this ellipse.( ) x ∣∣{ (x 2 /4) + (y 2 /9) = 1}y̌ 1.28 Imagine a rope wound around the earth’s equator so that it fits snugly (supposethat the earth is a sphere). How much extra rope must we add to raise the circleto a constant six feet off the ground?̌ 1.29 Verify that this map h: R 3 → R⎛ ⎞ ⎛ ⎞ ⎛ ⎞x x 3⎝y⎠ ↦→ ⎝y⎠ • ⎝−1⎠ = 3x − y − zz z −1is linear. Generalize.1.30 Show that every homomorphism from R 1 to R 1 acts via multiplication by ascalar. Conclude that every nontrivial linear transformation of R 1 is an isomorphism.Is that true for transformations of R 2 ? R n ?1.31 (a) Show that for any scalars a 1,1 , . . . , a m,n this map h: R n → R m is a homomorphism.⎛ ⎞ ⎛⎞x 1 a 1,1 x 1 + · · · + a 1,n x n⎜⎝⎟. ⎠ ↦→ ⎜⎝⎟. ⎠x n a m,1 x 1 + · · · + a m,n x n(b) Show that for each i, the i-th derivative operator d i /dx i is a linear transformationof P n . Conclude that for any scalars c k , . . . , c 0 this map is a lineartransformation of that space.f ↦→ dkdx f + c d k−1k k−1dx f + · · · + c dk−1 1dx f + c 0f1.32 Lemma 1.16 shows that a sum of linear functions is linear and that a scalarmultiple of a linear function is linear. Show also that a composition of linearfunctions is linear.̌ 1.33 Where f: V → W is linear, suppose that f(⃗v 1 ) = ⃗w 1 , . . . , f(⃗v n ) = ⃗w n for somevectors ⃗w 1 , . . . , ⃗w n from W.(a) If the set of ⃗w ’s is independent, must the set of ⃗v ’s also be independent?(b) If the set of ⃗v ’s is independent, must the set of ⃗w ’s also be independent?(c) If the set of ⃗w ’s spans W, must the set of ⃗v ’s span V?(d) If the set of ⃗v ’s spans V, must the set of ⃗w ’s span W?1.34 Generalize Example 1.15 by proving that for every appropriate domain andcodomain the matrix transpose map is linear. What are the appropriate domainsand codomains?