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Show that R3 is a vector space.

Question

Show that R3 is a vector space.

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Solution

To show that R^3 is a vector space, we need to verify that it satisfies the ten axioms of a vector space. The elements of R^3 are ordered triples of real numbers, which we can think of as vectors in three-dimensional space.

  1. Closure under addition: If u = (a, b, c) and v = (d, e, f) are in R^3, then u + v = (a + d, b + e, c + f) is also in R^3.

  2. Commutativity of addition: For any u = (a, b, c) and v = (d, e, f) in R^3, u + v = v + u.

  3. Associativity of addition: For any u, v, w in R^3, (u + v) + w = u + (v + w).

  4. Additive identity: The zero vector 0 = (0, 0, 0) is in R^3, and for any u in R^3, u + 0 = u.

  5. Additive inverse: For any u = (a, b, c) in R^3, there is a vector -u = (-a, -b, -c) in R^3 such that u + (-u) = 0.

  6. Closure under scalar multiplication: If u = (a, b, c) is in R^3 and k is a real number, then ku = (ka, kb, kc) is in R^3.

  7. Distributivity of scalar multiplication with respect to vector addition: For any u, v in R^3 and any scalar k, k(u + v) = ku + kv.

  8. Distributivity of scalar multiplication with respect to scalar addition: For any u in R^3 and any scalars k and l, (k + l)u = ku + lu.

  9. Compatibility of scalar multiplication with scalar multiplication: For any u in R^3 and any scalars k and l, (kl)u = k(lu).

  10. Scalar multiplication identity: For any u in R^3, 1u = u.

Since R^3 satisfies all ten axioms, it is a vector space.

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