Span and Linear Independence

Find a list of four distinct vectors in $\mathbf{F}^{3}$ whose span equals

Prove or give a counterexample: If $v_{1}, v_{2}, v_{3}, v_{4}$ spans $V$, then the list

also spans $V$.

Suppose $v_{1}, \ldots , v_{m}$ is a list of vectors in $V$. For $k \in \left\{ 1, \ldots , m\right\}$, let

Show that $\operatorname {span}\left(v_{1}, \ldots , v_{m}\right) = \operatorname {span}\left(w_{1}, \ldots , w_{m}\right)$.

Find a number $t$ such that

is not linearly independent in $\mathbb {R}^{3}$.

Show that the list (2,3,1), (1, -1,2), (7,3,c) is linearly dependent in $\mathbf{F}^{3}$ if and only if $c = 8$.

Suppose $v_{1}, v_{2}, v_{3}, v_{4}$ is linearly independent in $V$. Prove that the list

is also linearly independent.

Prove or give a counterexample: If $v_{1}, v_{2}, \ldots , v_{m}$ is a linearly independent list of vectors in $V$, then

is linearly independent.

Prove or give a counterexample: If $v_{1}, v_{2}, \ldots , v_{m}$ is a linearly independent list of vectors in $V$ and $\lambda \in \mathbf{F}$ with $\lambda \neq 0$, then $\lambda v_{1}, \lambda v_{2}, \ldots , \lambda v_{m}$ is linearly independent.

Prove or give a counterexample: If $v_{1}, \ldots , v_{m}$ and $w_{1}, \ldots , w_{m}$ are linearly independent lists of vectors in $V$, then the list $v_{1}+w_{1}, \ldots , v_{m}+w_{m}$ is linearly independent.

Suppose $v_{1}, \ldots , v_{m}$ is linearly independent in $V$ and $w \in V$. Prove that if $v_{1}+w, \ldots , v_{m}+w$ is linearly dependent, then $w \in \operatorname {span}\left(v_{1}, \ldots , v_{m}\right)$.

Suppose $v_{1}, \ldots , v_{m}$ is linearly independent in $V$ and $w \in V$. Show that

Suppose $v_{1}, \ldots , v_{m}$ is a list of vectors in $V$. For $k \in \left\{ 1, \ldots , m\right\}$, let

Show that the list $v_{1}, \ldots , v_{m}$ is linearly independent if and only if the list $w_{1}, \ldots , w_{m}$ is linearly independent.

Explain why there does not exist a list of six polynomials that is linearly independent in $\mathcal{P}_{4}(\mathbf{F})$.

Explain why no list of four polynomials spans $\mathcal{P}_{4}(\mathbf{F})$.

Prove that $V$ is infinite-dimensional if and only if there is a sequence $v_{1}, v_{2}, \ldots$ of vectors in $V$ such that $v_{1}, \ldots , v_{m}$ is linearly independent for every positive integer $m$.

Prove that $\mathbf{F}^{\infty }$ is infinite-dimensional.

Prove that the real vector space of all continuous real-valued functions on the interval $[0,1]$ is infinite-dimensional.

Suppose $p_{0}, p_{1}, \ldots , p_{m}$ are polynomials in $\mathcal{P}_{m}(\mathbf{F})$ such that $p_{k}(2) = 0$ for each $k \in \left\{ 0, \ldots , m\right\}$. Prove that $p_{0}, p_{1}, \ldots , p_{m}$ is not linearly independent in $\mathcal{P}_{m}(\mathbf{F})$.