Linear Algebra

Section II. Homomorphisms 189(d) the zero map Z: R 3 → R 4̌ 2.24 Find the nullity of each map.(a) h: R 5 → R 8 of rank five (b) h: P 3 → P 3 of rank one(c) h: R 6 → R 3 , an onto map (d) h: M 3×3 → M 3×3 , ontǒ 2.25 What is the null space of the differentiation transformation d/dx: P n → P n ?What is the null space of the second derivative, as a transformation of P n ? Thek-th derivative?2.26 Example 2.7 restates the first condition in the definition of homomorphism as‘the shadow of a sum is the sum of the shadows’. Restate the second condition inthe same style.2.27 For the homomorphism h: P 3 → P 3 given by h(a 0 + a 1 x + a 2 x 2 + a 3 x 3 ) =a 0 + (a 0 + a 1 )x + (a 2 + a 3 )x 3 find these.(a) N (h) (b) h −1 (2 − x 3 ) (c) h −1 (1 + x 2 )̌ 2.28 For the map f: R 2 → R given by( xf( ) = 2x + yy)sketch these inverse image sets: f −1 (−3), f −1 (0), and f −1 (1).̌ 2.29 Each of these transformations of P 3 is one-to-one. Find the inverse of each.(a) a 0 + a 1 x + a 2 x 2 + a 3 x 3 ↦→ a 0 + a 1 x + 2a 2 x 2 + 3a 3 x 3(b) a 0 + a 1 x + a 2 x 2 + a 3 x 3 ↦→ a 0 + a 2 x + a 1 x 2 + a 3 x 3(c) a 0 + a 1 x + a 2 x 2 + a 3 x 3 ↦→ a 1 + a 2 x + a 3 x 2 + a 0 x 3(d) a 0 +a 1 x+a 2 x 2 +a 3 x 3 ↦→ a 0 +(a 0 +a 1 )x+(a 0 +a 1 +a 2 )x 2 +(a 0 +a 1 +a 2 +a 3 )x 32.30 Describe the null space and range space of a transformation given by ⃗v ↦→ 2⃗v.2.31 List all pairs (rank(h), nullity(h)) that are possible for linear maps from R 5 toR 3 .2.32 Does the differentiation map d/dx: P n → P n have an inverse?̌ 2.33 Find the nullity of the map h: P n → R given bya 0 + a 1 x + · · · + a n x n ↦→∫ x=1x=0a 0 + a 1 x + · · · + a n x n dx.2.34 (a) Prove that a homomorphism is onto if and only if its rank equals thedimension of its codomain.(b) Conclude that a homomorphism between vector spaces with the same dimensionis one-to-one if and only if it is onto.2.35 Show that a linear map is one-to-one if and only if it preserves linear independence.2.36 Corollary 2.18 says that for there to be an onto homomorphism from a vectorspace V to a vector space W, it is necessary that the dimension of W be lessthan or equal to the dimension of V. Prove that this condition is also sufficient;use Theorem 1.9 to show that if the dimension of W is less than or equal to thedimension of V, then there is a homomorphism from V to W that is onto.̌ 2.37 Recall that the null space is a subset of the domain and the range space is asubset of the codomain. Are they necessarily distinct? Is there a homomorphismthat has a nontrivial intersection of its null space and its range space?2.38 Prove that the image of a span equals the span of the images. That is, whereh: V → W is linear, prove that if S is a subset of V then h([S]) equals [h(S)]. Thisgeneralizes Lemma 2.1 since it shows that if U is any subspace of V then its image{h(⃗u) ∣ ∣ ⃗u ∈ U} is a subspace of W, because the span of the set U is U.