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Exercise 5.1.6

Avg difficulty 3.0

In this exercise, we use scalars from the field Z5\mathbb{Z}_{5} of integers modulo 55 instead of real numbers (see Section an exercise given elsewhere in the book, “Fields”). Consider the vectors

u1=[143],u2=[023],u3=[213],u4=[310].\mathbf{u}_{1}=\left[\begin{array}{r}1\\ 4\\ 3\end{array}\right],\quad\mathbf{u}_{2}=\left[\begin{array}{r}0\\ 2\\ 3\end{array}\right],\quad\mathbf{u}_{3}=\left[\begin{array}{r}2\\ 1\\ 3\end{array}\right],\quad\mathbf{u}_{4}=\left[\begin{array}{r}3\\ 1\\ 0\end{array}\right].

Which of the following vectors are in span⁡{u1,…,u4}\operatorname{span}\left\{\mathbf{u}_{1},\ldots,\mathbf{u}_{4}\right\}? For each vector that is in the span, exhibit a linear combination of u1,…,u4\mathbf{u}_{1},\ldots,\mathbf{u}_{4} that equals this vector.

(a)x=[014],(b)y=[112],(c)z=[204].(a)\quad\mathbf{x}=\left[\begin{array}{r}0\\ 1\\ 4\end{array}\right],\qquad(b)\quad\mathbf{y}=\left[\begin{array}{r}1\\ 1\\ 2\end{array}\right],\qquad(c)\quad\mathbf{z}=\left[\begin{array}{r}2\\ 0\\ 4\end{array}\right].

Selinger, Matrix Theory and Linear Algebra, Spans, linear independence, and bases in Rn, Spans, Exercise 5.1.6

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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.