Let U be the set of all circles in R2 having center at the origin with radius r where r ≥ 0. Definethe sum of two circles of radius r1 and r2 to be a circles with radius r1 + r2. Also define the scalar multiple kof a circle with radius r1 to be a circle with radius |k|r1. Determine if U is a vector space.

Let U=x,y: x2+y2=r2 : ∀ x, y∈R, r≥0 u1=x1,y1∈ℝ2:x12+y12=r12, r1≥0u2=x2,y2∈ℝ2:x22+y22=r22,…

Answered: Let U be the set of all circles in R2…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

Let U be the set of all circles in R² having center at the origin with radius r where r ≥ 0. Define
the sum of two circles of radius r₁ and r₂ to be a circles with radius r₁ + r₂. Also define the scalar multiple k
of a circle with radius r₁ to be a circle with radius |k|r₁. Determine if U is a vector space.

Expert Solution

Step 1: Defining the property closed under addition and scalar multiple property

Let $U = \{(x, y) : x^{2} + y^{2} = r^{2} : \forall x, y \in R, r \geq 0\}$

$u_{1} = \{(x_{1}, y_{1}) \in ℝ^{2} : x_{1}^{2} + y_{1}^{2} = r_{1}^{2}, r_{1} \geq 0\} u_{2} = \{(x_{2}, y_{2}) \in ℝ^{2} : x_{2}^{2} + y_{2}^{2} = r_{2}^{2}, r_{2} \geq 0\} ∴ u_{1} + u_{1} = \{(x_{1} + x_{2}, y_{1} + y_{2}) \in ℝ^{2} : x_{1}^{2} + y_{1}^{2} + x_{2}^{2} + y_{2}^{2} = r_{1}^{2} + r_{2}^{2}, r_{1} + r_{2} \geq 0\} (i)$

$∴ u_{1} + u_{2} \in U$

Therefore closed under addition

$u_{1} = \{(x_{1}, y_{1}) \in ℝ^{2} : x_{1}^{2} + y_{1}^{2} = r_{1}^{2}\} λ u_{1} = \{(λ x_{1}, λ y_{1}) \in ℝ^{2} : {(λ x_{1})}^{2} + {(λ y_{1})}^{2} = {(|λ| r_{1})}^{2}\} (i i)$