(f) (4 points) The set of ordered pairs of real numbers V= | 1VER is a...
4 Let R2 be the set of all ordered pairs of real numbers equipped with the operations: addition defined by (21,02) (91, 92) = (21 41, 22 y2) and scalar multiplication defined by c(x1,22) = (cx1,Cx2), herece R is a scalar. Note that the operation addition here is non standard. Is R’ in this case a vector space ? (Justify your answer)
VECTOR SPACES LINEAR ALGEBRA Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on u = (u1, u2) and v = (v1, v2): u + v = (u1 + v1 + 1, u2 + v2 + 1), ku = (ku1, ku2) a) Show that (0,0) does not = 0 b) Show that (-1, -1) = 0 c) Show that axiom 5 holds by producing an ordered pair -u...
V01 (version 953): Let V be the set of all pairs (x,y) of real numbers together with the following operations: (x1, yı) © (C2, y2) = (x1 + 22,41 + y2) cº (x, y) = (Acc, 4cg). (a) Show that scalar multiplication distributes over scalar addition, that is: (c+d) 9 (z, 3) = c+ (x, y) #de (x, y). (b) Explain why V nonetheless is not a vector space.
Let V be R2, the set of all ordered pairs (x, y) of real numbers. Define an operation of "addition" by (u, v) @ (x, y) = (u + x +1, v + y + 1) for all (u, v) and (x, y) in V. Define an operation of "scalar multipli- cation" by a® (x, y) = (ax, ay) for all a E R and (x,y) E V Under the two operations the set V is not a vector space....
[2 marks] Let V be the set of all ordered pairs of real numbers (u1, uv) with uj > 0. Consider the following addition and scalar multiplication operations on u = (u1, u2) and v = (v1, v2): u + v = (421, uz + v2), ku = (kuq, kuz) If the set V with the above operations satisfies Axiom 5 of a vector space (i.e., the existence of a negative element), what would be the negative of the vector...
linear algebra
1. Determine whether the given set, along with the specified operations of addition and scalar multiplication, is a vector space (over R). If it is not, list all of the axioms that fail to hold. a The set of all vectors in R2 of the form , with the usual vector addition and scalar multiplication b) R2 with the usual scalar multiplication but addition defined by 31+21 y1 y2 c) The set of all positive real numbers, with...
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Example. As another special case of examples we may regard the set R of all of n umber vector 1.4.6. Example. Yet another al l the vector space M of mx matrices of members of where m - NI. We will use M. horthand for M F ) and M. for M.(R) 1.4.9. Exercise. Let be the total real numbers. Define an operation of addition by y the maximum of u and y for...
Problem #9: Let V be the set of all ordered pairs of real numbers (uj,u) with up > 0. Consider the following addition and scalar multiplication operations on u = (U1, u) and v = (v1, v2): u + v = (U1 + V1 +4,5u2v2), ku = (kuj, kuz) Use the above operations for the following parts. (a) Compute u + v for u =(4,4) and v = (3,2). (b) If the set V satisfies Axiom 4 of a vector...
1. [4-+6+6-16 points Let /°0 denote the vector space of bounded sequences of real numbers, with addition and scalar multiplication defined componentwise. Define a norm Il on by Il xl = suplx! < oo where x = (x1,x2, 23, . .. ) iEN (a) Prove that is complete with respect to the norm | . (b) Consider the following subspaces of 1o i) c-the space of convergent sequences; (i) co-the space of sequences converging to 0; (iii) coo- the space...
2. On subspaces of C(-1,1) Let V C(-1,1) be the vector space of all continuous real valued functions on on the interval (-1, 1), with usual addition and scalar multiplication. (a) Verify, if the set W-f eV: f(0)-0is a subspace of V or not? (b) Verify, if the set W-Uev f(0) 1 is a subspace of V or not? (c) Verify, if the set W-İfEV:f(x)-0V-2-z is a subspace of V or not? 1b) PrtScn Home FS F6 F7 F8 5