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Exercise 5.2.12

Avg difficulty 3.0

In this exercise, we use scalars from the field Z3\mathbb{Z}_{3} of integers modulo 33 instead of real numbers (see Section an exercise given elsewhere in the book, “Fields”). Use the extended casting-out algorithm to determine which of the following vectors are redundant. If there are redundant vectors, write each of them as a linear combination of previous vectors.

u1=[210],u2=[112],u3=[202],u4=[011].\mathbf{u}_{1}=\left[\begin{array}{r}2\\ 1\\ 0\end{array}\right],\quad\mathbf{u}_{2}=\left[\begin{array}{r}1\\ 1\\ 2\end{array}\right],\quad\mathbf{u}_{3}=\left[\begin{array}{r}2\\ 0\\ 2\end{array}\right],\quad\mathbf{u}_{4}=\left[\begin{array}{r}0\\ 1\\ 1\end{array}\right].

Selinger, Matrix Theory and Linear Algebra, Spans, linear independence, and bases in Rn, Linear independence, Exercise 5.2.12

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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.