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Exercise 9.2.8

Avg difficulty 3.0

Consider polynomials

\left\{a_{i}x^{3}+b_{i}x^{2}+c_{i}x+d_{i}\mid i=1,2,3,4\right\}.

Show that this collection of polynomials is linearly independent if and only if

\left[\begin{array}{cccc}a_{1}&b_{1}&c_{1}&d_{1}\\ a_{2}&b_{2}&c_{2}&d_{2}\\ a_{3}&b_{3}&c_{3}&d_{3}\\ a_{4}&b_{4}&c_{4}&d_{4}\end{array}\right]

is an invertible matrix.

Selinger, Matrix Theory and Linear Algebra, Vector spaces, Linear combinations, span, and linear independence, Exercise 9.2.8

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.