Spans

Consider the vectors

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

Describe the span of the vectors $\mathbf{u}=\left[\begin{array}{r}1\\ 0\\ 2\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{r}2\\ -1\\ 1\end{array}\right]$ in $\mathbb{R}^{3}$.

Describe the span of the following vectors in $\mathbb{R}^{4}$:

Suppose $\left\{\mathbf{u}_{1},\ldots,\mathbf{u}_{k}\right\}$ is a set of vectors from $\mathbb{R}^{n}$. Show that $\mathbf{0}\in\operatorname{span}\left\{\mathbf{u}_{1},\ldots,\mathbf{u}_{k}\right\}$.

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

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