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Problem 2

Avg difficulty 3.0

Suppose VV is a vector space over the field FF with basis {v1,v2,v3}\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\}. Let

u1\displaystyle\mathbf{u}_{1} =v1,\displaystyle=\mathbf{v}_{1}, u2\displaystyle\mathbf{u}_{2} =v1+v2, and\displaystyle=\mathbf{v}_{1}+\mathbf{v}_{2},\text{ and} u3\displaystyle\mathbf{u}_{3} =v1+v2+v3.\displaystyle=\mathbf{v}_{1}+\mathbf{v}_{2}+\mathbf{v}_{3}.

Prove that {u1,u2,u3}\{\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{3}\} is also a basis for VV.

Kan, A Complete Introduction to Abstract Algebra, Fields, Vector Spaces, Problem 2

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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.