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Exercise 6

Avg difficulty 3.0

For the vector space

V={f(x)∈Q[x]∣deg⁡f(x)≤2}∪{0}V=\{f(x)\in\mathbb{Q}[x]\mid\deg f(x)\leq 2\}\cup\{0\}

over Q\mathbb{Q}, which of the following sets of vectors of VV is/are linearly independent?

  1. S={1,x+2}S=\{1,x+2\}

  2. T={1,x+2,x2+2x+3}T=\{1,x+2,x^{2}+2x+3\}

  3. U={1,x+2,x2+2x+3,x2−2x−3}U=\{1,x+2,x^{2}+2x+3,x^{2}-2x-3\}

Kan, A Complete Introduction to Abstract Algebra, Fields, Vector Spaces, Exercise 6

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Adapted from A Complete Introduction to Abstract Algebra © 2022–2026 Kan Onn Kit. Licensed under CC BY-SA 4.0. Source: A Complete Introduction to Abstract Algebra. Imported at revision a559e8b.