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Exercise 5.1.1

Avg difficulty 3.0

Consider the vectors

u1=[11−2],u2=[12−2],u3=[1−3−2],u4=[−112].\mathbf{u}_{1}=\left[\begin{array}{r}1\\ 1\\ -2\end{array}\right],\quad\mathbf{u}_{2}=\left[\begin{array}{r}1\\ 2\\ -2\end{array}\right],\quad\mathbf{u}_{3}=\left[\begin{array}{r}1\\ -3\\ -2\end{array}\right],\quad\mathbf{u}_{4}=\left[\begin{array}{r}-1\\ 1\\ 2\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=[12−1],(b)y=[2−3−4],(c)z=[15−2].(a)\quad\mathbf{x}=\left[\begin{array}{r}1\\ 2\\ -1\end{array}\right],\qquad(b)\quad\mathbf{y}=\left[\begin{array}{r}2\\ -3\\ -4\end{array}\right],\qquad(c)\quad\mathbf{z}=\left[\begin{array}{r}1\\ 5\\ -2\end{array}\right].

Selinger, Matrix Theory and Linear Algebra, Spans, linear independence, and bases in Rn, Spans, Exercise 5.1.1

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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.