Linear Algebra

186 Chapter Three. Maps Between Spacesone-to-one, each inverse image is a single point and the map is an isomorphismbetween the domain and the range.Exerciseš 2.22 Let h: P 3 → P 4 be given by p(x) ↦→ x · p(x). Which of these are in thenullspace? Which are in the rangespace?(a) x 3 (b) 0 (c) 7 (d) 12x − 0.5x 3 (e) 1 + 3x 2 − x 3̌ 2.23 Find the nullspace, nullity, rangespace, and rank of each map.(a) h: R 2 → P 3 given by(ab)↦→ a + ax + ax 2(b) h: M 2×2 → R given by( )a b↦→ a + dc d(c) h: M 2×2 → P 2 given by(a(d) the zero map Z : R 3 → R 4c)b↦→ a + b + c + dx 2ď 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 nullspace of the differentiation transformation d/dx: P n → P n ?What is the nullspace 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 1x + a 2x 2 + a 3x 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(xy)f() = 2x + ysketch these inverse image sets: f −1 (−3), f −1 (0), and f −1 (1).̌ 2.29 Each of these transformations of P 3 is nonsingular. Find the inverse functionof each.(a) a 0 + a 1x + a 2x 2 + a 3x 3 ↦→ a 0 + a 1x + 2a 2x 2 + 3a 3x 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 1x + a 2x 2 + a 3x 3 ↦→ a 1 + a 2x + a 3x 2 + a 0x 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 nullspace and rangespace of a transformation given by ⃗v ↦→ 2⃗v.2.31 List all pairs (rank(h), nullity(h)) that are possible for linear maps from R 5to R 3 .2.32 Does the differentiation map d/dx: P n → P n have an inverse?