Linear Algebra

218 Chapter Three. Maps Between Spaces(a)(c)̌ 2.15 Where( ) ( )3 1 0 5(b)−4 2 0 0.5( ) ⎛ ⎞1 0 52 −7⎝−1 1 1⎠7 43 8 4A =( ) 1 −12 0( ) ⎛ ⎞−1 −11 1 −1⎝23 1 1⎠4 0 33 1 1( ) ( )5 2 −1 2(d)3 1 3 −5B =( ) 5 24 4compute or state ‘not defined’.(a) AB (b) (AB)C (c) BC (d) A(BC)C =( −2) 3−4 12.16 Which products are defined?(a) 3×2 times 2×3 (b) 2×3 times 3×2 (c) 2×2 times 3×3(d) 3×3 times 2×2̌ 2.17 Give the size of the product or state “not defined”.(a) a 2×3 matrix times a 3×1 matrix(b) a 1×12 matrix times a 12×1 matrix(c) a 2×3 matrix times a 2×1 matrix(d) a 2×2 matrix times a 2×2 matrix̌ 2.18 Find the system of equations resulting from starting withh 1,1 x 1 + h 1,2 x 2 + h 1,3 x 3 = d 1h 2,1 x 1 + h 2,2 x 2 + h 2,3 x 3 = d 2and making this change of variable (i.e., substitution).x 1 = g 1,1 y 1 + g 1,2 y 2x 2 = g 2,1 y 1 + g 2,2 y 2x 3 = g 3,1 y 1 + g 3,2 y 22.19 As Definition 2.3 points out, the matrix product operation generalizes the dotproduct. Is the dot product of a 1×n row vector and a n×1 column vector thesame as their matrix-multiplicative product?̌ 2.20 Represent the derivative map on P n with respect to B, B where B is the naturalbasis 〈1, x, . . . , x n 〉. Show that the product of this matrix with itself is defined;what the map does it represent?2.21 [Cleary] Match each type of matrix with all these descriptions that could fit:(i) can be multiplied by its transpose to make a 1×1 matrix, (ii) is similar to the3×3 matrix of all zeros, (iii) can represent a linear map from R 3 to R 2 that is notonto, (iv) can represent an isomorphism from R 3 to P 2 .(a) a 2×3 matrix whose rank is 1(b) a 3×3 matrix that is nonsingular(c) a 2×2 matrix that is singular(d) an n×1 column vector2.22 Show that composition of linear transformations on R 1 is commutative. Is thistrue for any one-dimensional space?2.23 Why is matrix multiplication not defined as entry-wise multiplication? Thatwould be easier, and commutative too.̌ 2.24 (a) Prove that H p H q = H p+q and (H p ) q = H pq for positive integers p, q.(b) Prove that (rH) p = r p · H p for any positive integer p and scalar r ∈ R.̌ 2.25 (a) How does matrix multiplication interact with scalar multiplication: isr(GH) = (rG)H? Is G(rH) = r(GH)?