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?

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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?

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(u ℗ v) ¤ V
u v = vu
(u + v) w=uⒸ (v © w)
u +0=u
u + (-u) = 0
(@sh
r*u € V
(r+s) *u = r*us*u
r* (u + v) = r *u Ⓒr*v
r* (s*u) = (rs) * u
1 *u = u
Transcribed Image Text:1 2 3 4 5 6 7 8 10 (u ℗ v) ¤ V u v = vu (u + v) w=uⒸ (v © w) u +0=u u + (-u) = 0 (@sh r*u € V (r+s) *u = r*us*u r* (u + v) = r *u Ⓒr*v r* (s*u) = (rs) * u 1 *u = u
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