Let V denote the set of all differentiablc real-valued functions defined on the real

Label the following statements as true or false. (a) Every vector space contains a zero vector. (b) A vector space may have more than one zero vector. (c) In any vector space, ax = bx implies that a = 6. (d) In any vector space, ax = ay implies that x = y. (e) A vector in F" may be regarded as a matrix in M.xi(F). (f) An m x n matrix has m columns and n rows. (g) In P(F). only polynomials of the same degree may be added. (h) If / and o are polynomials of degree n, then / + g is a polynomial of degree n. (i) If / is a polynomial of degree n and c is a nonzero scalar, then cf is a polynomial of degree n. Sec. 1.2 Vector Spaces 13 (j) A nonzero scalar of F may be considered to be a polynomial in P(F) having degree zero, (k) Two functions in F(S, F) are equal if and only if they have the same value at each element of S.

Write the zero vector of 3. If "3x4 (F).

If "3x4 (F). M = 1 2 3 4 5 6 what are Mia,M2i, and M22?

Perform the indicated operations. w <\ I 1 (d) (e) (2.T4 - 7x3 + Ax + 3) + (8x3 + 2x2 - 6x + 7) (f) (-3a;3 + 7a;2 + 8a - 6) + (2a:3 - 8ar + 10) (g) 5(2.r7 - 6x4 + 8x2 - 3a) (h) 3(.x5 - 2a:3 + 4a; + 2)

Richard Card ("Effects of Beaver on Trout in Sagehen Creek. California," ./. Wildlife Management. 25, 221-242) reports the following number of trout, having crossed beaver dams in Sagehen Creek.

At the end of May, a furniture1 store had the following inventory. Early MediterAmerican Spanish ranean Danish Living room suites Bedroom suites Dining room suites 1 5 3 2 1 1 ] I 2 3 4 6 5 (i 1 3 2 0 1 1 3 2 ) 3 Record these data as a 3x 4 matrix M. To prepare for its June sale, the store decided to double its inventory on each of the items listed in the preceding table. Assuming that none of the present stock is sold until the additional furniture arrives, verify that the inventory on hand after the order is filled is described by the matrix 2M. If the inventory at the end of June is described by the matrix

Let 5 = {(). 1} and F - R. In F(S, R), show that. / = g and / -t g = h. where f(t) = 2t + 1. g(t) = 1 + At - It2 , and h(t) = 5' + 1.

In any vector space1 V, show that, (a + b)(x + y) = ax + ay + bx + by for any x.y V and any o,6 F.

Prove Corollaries 1 and 2 of Theorem 1.1 and Theorem 1.2(c).

Let V denote the set of all differentiablc real-valued functions defined on the real line. Prove that V is a vector space with the operations of addition and scalar multiplication defined in Example 3.

Let V = {()} consist of a single vector 0 and define 0 + 0 = 0 and cO = 0 for each scalar c in F. Prove that V is a vector space over F. (V is called the zero vector space.)

A real-valued function / defined on the real line is called an even function if /(/,) = f(t) for each real number t. Prove that the set. of even functions defined on the real line with the operations of addition and scalar multiplication defined in Example 3 is a vector space1

Let V denote the set of ordered pairs of real numbers. If (01,02) and (6], 62) are elements of V and c R, define (a 1,

Let V = {(c/.i, a2..... on ): c/.( C for i = 1,2,... n}; so V is a vector space over C by Example 1. Is V a. vector space over the field of real numbers with the operations of coordinatewise addition and multiplication?

Let V = {(a.]. 0'2: an): aL Riori 1,2, ...n}; so V is a vector space over R by Example 1. Is V a vector space over the field of complex numbers with the operations of coordinatewise addition and multiplication?

Let V denote the set of all m x n matrices with real entries; so V is a vector space over R by Example 2. Let. F be the field of rational numbers. Is V a vector space over F with the usual definitions of matrix addition and scalar multiplication?

Let V = {(01,02): 01,02 F}, where F is a field. Define addition of elements of V coordinatewise. and for c F and (01,02) V, define c(ai,o2) = (c/.|,0). Is V a vector space over F with these operations? Justify your answer.

Let V = {(01,02): oi,a 2 R}. For (ai,a 2 ),(6i,62 ) V and c R. define (a,\.a2) + (6].62) = (ai + 26i.a-2 + 3b2) and c{a.\,a2) = (cai,ca2)- Is V a vector space over R with these operations? Justify your answer.

Let V = {(01,02): 01,02 R}- Define addition of elements of V coordinatewise, and for (01,02) in V and c R, define ;o, 0) if c = 0 cai, 1 it c / 0. Is V a vector space over R with these operations? Justify your answer

Let V be the set of sequences {an} of real numbers. (See Example 5 for the definition of a sequence.) For {}, {6n} V and any real number t. define {<>} + {6n} - {. + 6} and t{a} = {tan}. Prove that, with these operations. V is a vector space over R.

Let V and W be vector spaces over a field F. Let Z = {(v,w): v Vandv/; W}. Prove that Z is a vector space over F with the operations (vi,wi) + (v2.w2) = (ci + u2, w\ + w2) ;md c(v\,Wi) = (cvi,cw1).