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1. Give a subset of R2 with standard operations that is (i) a vector space (show all axioms) (ii) not a vector space (give a counter example) THE TEN VECTOR SPACE AXIOMS 1. If u and v are objects in V, then u + v is in V. 2. u + v = v + u 3. u+ (v +w) = (u+ v) + w 4. There is an object 0 in V, called a zero vector for V, such that 0 + u = u + 0 = u for all u in V. 5. For each u in V, there is an object -u in V, called a negative of u, such that u+ (-u) 6. If k is any scalar and u in any object in V, then ku is in V. 7. k(u+ v) 8. (k+ m)u 9. К(mи) — (km)u (-u) + u = 0. ku + kv ku + mu 10. lu=u

1. Give a subset of R2 with standard operations that is (i) a vector space (show all axioms) (ii) not a vector space (give a counter example) THE TEN VECTOR SPACE AXIOMS 1. If u and v are objects in V, then u + v is in V. 2. u + v = v + u 3. u+ (v +w) = (u+ v) + w 4. There is an object 0 in V, called a zero vector for V, such that 0 + u = u + 0 = u for all u in V. 5. For each u in V, there is an object -u in V, called a negative of u, such that u+ (-u) 6. If k is any scalar and u in any object in V, then ku is in V. 7. k(u+ v) 8. (k+ m)u 9. К(mи) — (km)u (-u) + u = 0. ku + kv ku + mu 10. lu=u

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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1. Give a subset of R2 with standard operations that is (i) a vector space (show all axioms) (ii) not a vector space (give a counter example)
Transcribed Image Text:1. Give a subset of R2 with standard operations that is (i) a vector space (show all axioms) (ii) not a vector space (give a counter example)
THE TEN VECTOR SPACE AXIOMS 1. If u and v are objects in V, then u + v is in V. 2. u + v = v + u 3. u+ (v +w) = (u+ v) + w 4. There is an object 0 in V, called a zero vector for V, such that 0 + u = u + 0 = u for all u in V. 5. For each u in V, there is an object -u in V, called a negative of u, such that u+ (-u) 6. If k is any scalar and u in any object in V, then ku is in V. 7. k(u+ v) 8. (k+ m)u 9. К(mи) — (km)u (-u) + u = 0. ku + kv ku + mu 10. lu=u
Transcribed Image Text:THE TEN VECTOR SPACE AXIOMS 1. If u and v are objects in V, then u + v is in V. 2. u + v = v + u 3. u+ (v +w) = (u+ v) + w 4. There is an object 0 in V, called a zero vector for V, such that 0 + u = u + 0 = u for all u in V. 5. For each u in V, there is an object -u in V, called a negative of u, such that u+ (-u) 6. If k is any scalar and u in any object in V, then ku is in V. 7. k(u+ v) 8. (k+ m)u 9. К(mи) — (km)u (-u) + u = 0. ku + kv ku + mu 10. lu=u
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