Linear Algebra

176 Chapter Three. Maps Between Spaceš 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.(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 thenh(⃗v) = r · ⃗v for 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 .( )x x/2↦→y)y/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 be added to raise thecircle to a constant six feet off the ground?̌ 1.29 Verify that this map h: R 3 → R( ) xyz↦→( xyz) ( 3−1−1)= 3x − y − zis 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.d k−1f ↦→ dkdx f + c k k−1dx f + · · · + dk−1 c1dx f + c0f1.32 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 forsome vectors ⃗w 1, . . . , ⃗w n from W .(a) If the set of ⃗w ’s is independent, must the set of ⃗v ’s also be independent?