Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN: 9781305658004
Author: Ron Larson
Publisher: Cengage Learning
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Chapter A, Problem 24E
To determine
To prove:
The zero
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Using the 10 properties:(a) Show that the set of strictly positive reals: R+ = {a E R | a > 0}, provided with the operations defined below is a vector space:- Addition: a + b = ab for all a, b E R+- Multiplication by a scalar: k*a=ak, for all k E R and a E R+
(b) Why is the set of positive reals { a E R | a ⩾ 0}, endowed with the same operations, not a vector space?
Use proof by contradiction to prove the statement.In a vector space, the zero vector is unique.
#Consider the number line R. Let a
"vector" be a real number and the scalars a &
b in properties (e), (f), (g) on pg 223 can only
be rational numbers. Show that R is a
vector space over the rational numbers Q,
using standard addition and multiplication of
real numbers.
Chapter A Solutions
Elementary Linear Algebra (MindTap Course List)
Ch. A - Using Mathematical Induction In Exercises 1-4, use...Ch. A - Prob. 2ECh. A - Prob. 3ECh. A - Prob. 4ECh. A - Prob. 5ECh. A - Prob. 6ECh. A - Prob. 7ECh. A - Prob. 8ECh. A - Prob. 9ECh. A - Prob. 10E
Ch. A - Prob. 11ECh. A - Prob. 12ECh. A - Prob. 13ECh. A - Prob. 14ECh. A - Using Proof by Contradiction In Exercises 1526,...Ch. A - Prob. 16ECh. A - Prob. 17ECh. A - Prob. 18ECh. A - Prob. 19ECh. A - Prob. 20ECh. A - Prob. 21ECh. A - Prob. 22ECh. A - Prob. 23ECh. A - Prob. 24ECh. A - Prob. 25ECh. A - Prob. 26ECh. A - Prob. 27ECh. A - Prob. 28ECh. A - Prob. 29ECh. A - Prob. 30ECh. A - Prob. 31ECh. A - Prob. 32ECh. A - Prob. 33E
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- ProofProve in full detail that the set {(x,2x):xisarealnumber}, with the standard operations in R2, is a vector space.arrow_forwardProofProve in full detail that M2,2, with the standard operations, is a vector space.arrow_forwardProof Prove that u+v=u+v if and only if u and v have the same direction.arrow_forward
- Proof In Exercises 6568, complete the proof of the remaining properties of theorem 4.3 by supplying the justification for each step. Use the properties of vector addition and scalar multiplication from theorem 4.2. Property 6: (v)=v (v)+(v)=0andv+(v)=0a.(v)+(v)=v+(v)b.(v)+(v)+v=v+(v)+vc.(v)+((v)+v)=v+((v)+v)d. (v)+0=v+0e.(v)=vf.arrow_forwardProof Prove that the scalar 1 is the identity for scalar multiplication: 1A=A.arrow_forwardLinear Algebraarrow_forward
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