Linear combinations, span, and linear independence

Let $V$ be a vector space and suppose $\left\{\mathbf{u}_{1},\ldots,\mathbf{u}_{k}\right\}$ is a set of vectors in $V$. Show that $\mathbf{0}$ is in $\operatorname{span}\left\{\mathbf{u}_{1},\ldots,\mathbf{u}_{k}\right\}$.

Let $K$ be a field, and consider the vector space $\mathbf{Seq}_{K}$ of infinite sequences of scalars. A sequence $a=(a_{i})_{i\in\mathbb{N}}$ is called finitely supported if all but finitely many elements of the sequence are zero. In other words, $a$ is finitely supported if there exists some $N\in\mathbb{N}$ such that $a_{k}=0$ for all $k\geq N$. Let $e^{0},e^{1},e^{2},\ldots$ be the sequences from Example 9.19. Show that $a\in\operatorname{span}\left\{e^{0},e^{1},e^{2},\ldots\right\}$ if and only if $a$ is finitely supported.

Consider polynomials

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

is an invertible matrix.

Assume $\mathbf{u},\mathbf{v},\mathbf{w}$ are linearly independent elements of some vector space $V$. Consider the set of vectors

Determine whether $R$ is linearly independent.