Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and

Consider the vectors x1 = (8, 6)T and x2 = (4, 1)T in R2. (a) Determine the length of each vector. (b) Let x3 = x1 + x2. Determine the length of x3. How does this length compare with the sum of the lengths of x1 and x2? (c) Draw a graph illustrating how x3 can be constructed geometrically using x1 and x2. Use this graph to give a geometrical interpretation of your answer to the question in part (b).

Repeat Exercise 1 for the vectors x1 = (2, 1)T and x2 = (6, 3)T .

Let C be the set of complex numbers. Define addition on C by (a + bi) + (c + di) = (a + c) + (b + d)i and define scalar multiplication by (a + bi) = a + bi for all real numbers . Show that C is a vector space with these operations.

Show that Rmn, together with the usual addition and scalar multiplication of matrices, satisfies the eight axioms of a vector space.

Show that C[a, b], together with the usual scalar multiplication and addition of functions, satisfies the eight axioms of a vector space.

Let P be the set of all polynomials. Show that P, together with the usual addition and scalar multiplication of functions, forms a vector space.

Show that the element 0 in a vector space is unique.

Let x, y, and z be vectors in a vector space V. Prove that if x + y = x + z then y = z.

Let V be a vector space and let x V. Show that (a) 0 = 0 for each scalar . (b) if x = 0, then either = 0 or x = 0. 1

Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition on S by (x1, x2) = (x1, x2) (x1, x2) (y1, y2) = (x1 + y1, 0) We use the symbol to denote the addition operation for this system in order to avoid confusion with the usual addition x + y of row vectors. Show that S, together with the ordinary scalar multiplication and the addition operation , is not a vector space. Which of the eight axioms fail to hold? 1

Let V be the set of all ordered pairs of real numbers with addition defined by (x1, x2) + (y1, y2) = (x1 + y1, x2 + y2) and scalar multiplication defined by (x1, x2) = (x1, x2) Scalar multiplication for this system is defined in an unusual way, and consequently we use the symbol to avoid confusion with the ordinary scalar multiplication of row vectors. Is V a vector space with these operations? Justify your answer. 1

Let R + denote the set of positive real numbers. Define the operation of scalar multiplication, denoted , by x = x for each x R + and for any real number . Define the operation of addition, denoted , by x y = x y for all x, y R + Thus, for this system, the scalar product of 3 times 1 2 is given by 3 1 2 = _ 1 2 _3 = 8 and the sum of 2 and 5 is given by 2 5 = 2 5 = 10 Is R + a vector space with these operations? Prove your answer. 1

Let R denote the set of real numbers. Define scalar multiplication by x = x (the usual multiplication of real numbers) and define addition, denoted , by x y = max(x, y) (the maximum of the two numbers) Is R a vector space with these operations? Prove your answer. 1

Let Z denote the set of all integers with addition defined in the usual way, and define scalar multiplication, denoted , by k = [[]] k for all k Z where [[]] denotes the greatest integer less than or equal to . For example, 2.25 4 = [[2.25]] 4 = 2 4 = 8 Show that Z, together with these operations, is not a vector space. Which axioms fail to hold? 1

Let S denote the set of all infinite sequences of real numbers with scalar multiplication and addition defined by {an} = {an} {an} + {bn} = {an + bn} Show that S is a vector space. 1

We can define a one-to-one correspondence between the elements of Pn and Rn by p(x) = a1 + a2x + +anxn1 (a1, . . . , an)T = a Show that if p a and q b, then (a) p a for any scalar . (b) p + q a + b. [In general, two vector spaces are said to be isomorphic if their elements can be put into a one-to-one correspondence that is preserved under scalar multiplication and addition as in (a) and (b).]