# Vector Spaces and Subspaces

2. Use Theorem 5.2.1 to determine which of the following are subspaces of R3.

Thm 5.2.1: If W is a set of one or more vectors from a vector space V, then W is a subspace of V if and only if the following conditions hold.

(a) If u and v are vectors in W, then u + v is in W.

(b) If k is any scalar and u is any vector in W, then ku is in W.

a) all vectors of the form (a, 0, 0)

b) all vectors of the form (a, 1, 1)

c) all vectors of the form (a, b, c), where b = a + c

d) all vectors of the form (a, b, c), where b = a + c + 1

e) all vectors of the form (a, b, 0)

18. Use Theorem 5.2.1 to determine which of the following are subspaces of P3.

Thm 5.2.1: If W is a set of one or more vectors from a vector space V, then W is a subspace of V if and only if the following conditions hold.

(c) If u and v are vectors in W, then u + v is in W.

(d) If k is any scalar and u is any vector in W, then ku is in W.

a) all polynomials a0 + a1x +a2x2 + a3x3 for which a0 = 0

b) all polynomials a0 + a1x +a2x2 + a3x3 for which a0 + a1 +a2 + a3 = 0

c) all polynomials a0 + a1x +a2x2 + a3x3 for which a0, a1, a2, a3 are integers

d) all polynomials of the form a0 + a1x, where a0 and a1 are real numbers

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#### Solution Summary

Vector spaces and subspaces are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.