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lem 6. Define new addition and scalar multiplication operations and Ⓒ on R² by (x, y) + (z, w) = (x+z,y+w), ao (x, y) = (ax, y), for all (x, y), (z, w) ≤ R² and a € R. Does R2 with these operations form a vector space over R? Justify your answer.

lem 6. Define new addition and scalar multiplication operations and Ⓒ on R² by (x, y) + (z, w) = (x+z,y+w), ao (x, y) = (ax, y), for all (x, y), (z, w) ≤ R² and a € R. Does R2 with these operations form a vector space over R? Justify your answer.

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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Problem 6. Define new addition and scalar multiplication operations and Ⓒ on R² by (x, y) + (z, w) = (x+z,y+w), a (x, y) = (ax, y), for all (x, y), (z, w) ≤ R² and a E space over R? Justify your answer. R. Does R2 with these operations form a vector
Transcribed Image Text:Problem 6. Define new addition and scalar multiplication operations and Ⓒ on R² by (x, y) + (z, w) = (x+z,y+w), a (x, y) = (ax, y), for all (x, y), (z, w) ≤ R² and a E space over R? Justify your answer. R. Does R2 with these operations form a vector
Expert Solution
Step 1

We have given that addition and scalar multiplication operations ⊕ and ⊙ on ℝ2 with all x, y, z, w∈ℝ2 and α∈ℝ.

x, y⊕z, w=x+z, y+wα⊙x, y=αx, y

We need to show that ℝ2 with respect to these operations form a vector space over ℝ.

 

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