Math 110 Homework Assignment 22 due date: Mar. 31, 2017 1. In each of the following cases, either prove that the given subset W is a subspace of V , or show why it is not a subspace of V . (a) V = C (R), W is the set of functions f V for which lim f (x) = 0. x (b) V = C (R), W is the set of functions f V for which f (4) = 1. (c) V = M22 (R), W is the set of matrices whose square is the zero matrix (i.e., those matrices A with A2 the zero matrix). (d) V = R , W is the subset of vectors (x1 , x2 , x3 , . . .) for which x3 = x2 + x1 , x4 = x3 + x2 , x5 = x4 + x3 , . . . , and in general xn+2 = xn+1 + xn for all n 1. 2. In each of the following cases, either prove that the vectors are linearly independent, or show that they are linearly dependent. (a) The vectors v1 = (1, 2) and v2 = (3, 6) in W2 . (b) The vectors v1 = ln(x2 + 1), v2 = ln(x4 + 4x2 + 3), and v3 = ln(x2 + 3) in C (R). (c) The vectors v1 = ex , v2 = e3x , and v3 = cos(x) in C (R). 3. In each of the following cases, either prove that the given rule is a linear transformation, or show why it isn't. (a) T : R R , T is the rule \"shift to the right\": T (x1 , x2 , x3 , . . .) = (0, x1 , x2 , x3 , . . .). (b) T : M23 (R) R2 , T sends the matrix M to M w ~ where w ~ = (1, 2, 3) (the output is a vector in R2 ). (c) T : W2 R2 , T (x, y) = (x, y). (d) T : W2 R2 , T (x, y) = (ln(x), ln(y)). Z 3 (e) T : C (R) R, T (f ) = f sin(x) dx. 1 1 The vector spaces used on the previous page are C (R): The vector space of functions from R to R with infinitely many derivatives. Addition and scalar multiplication are addition and scalar multiplication of functions. R : The set of infinite sequences (x1 , x2 , x3 , . . .) of numbers in R, where addition and scalar multiplication are coordinate-wise. Mmn (R): The set of m n matrices with entries in R, where addition is addition of matrices and scalar multiplication is multiplying all entries of the matrix by that number. W2 : The following \"weird\" vector space: W2 is the set of all pairs (x, y) with both x > 0 and y > 0 real numbers, with addition defined as multiplication of the coordinates: (x1 , y1) + (x2 , y2 ) = (x1 x2 , y1 y1 ) and scalar multiplication defined as exponentiation of the coordinates: c (x, y) = (xc , y c ), for any (x1 , y1), (x2 , y2) W2 , c R. Reminder: The zero vector in W2 is (1, 1). All of these vector spaces are vector spaces over R. 2