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by Problem 2. Consider the set V = R2 with addition and scalar multiplication defined (Y₁, Y2) = (x₁ + x2, Y₁ + y2), (x₁, x₂) = (ax₁, x₂). Is V a vector space with these operations? Justify your answer. (x1, x2) a

by Problem 2. Consider the set V = R2 with addition and scalar multiplication defined (Y₁, Y2) = (x₁ + x2, Y₁ + y2), (x₁, x₂) = (ax₁, x₂). Is V a vector space with these operations? Justify your answer. (x1, x2) a

Advanced Engineering Mathematics
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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by Problem 2. Consider the set V = R² with addition and scalar multiplication defined (X1, X2) ℗ (Y1, Y2) = (X1 + X2, Y₁ + y2), a (x₁, x₂) = (ax₁, x₂). Is V a vector space with these operations? Justify your answer.
Transcribed Image Text:by Problem 2. Consider the set V = R² with addition and scalar multiplication defined (X1, X2) ℗ (Y1, Y2) = (X1 + X2, Y₁ + y2), a (x₁, x₂) = (ax₁, x₂). Is V a vector space with these operations? Justify your answer.
Expert Solution
Step 1

Introduction:

A set X with the unique element 0 and the following three operations defines a real vector space.

  • When given two components in X, x and y, one can add the two together to create the sum, which is likewise an element in X.
  • An element of X can be used to create the inverse, or -x, which is also an element of X.
  • A real number c and an element of X, represented by the product cx, can be created via scalar multiplication.
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