Proof of Theorem DLDS

(⇒) Suppose that $S$ is linearly dependent, so there exists a nontrivial relation of linear dependence by Definition LICV. That is, there are scalars, $\alpha_i$, $1\leq i\leq n$, which are not all zero, such that \begin{equation*} \lincombo{\alpha}{u}{n}=\zerovector. \end{equation*} Since the $\alpha_i$ cannot all be zero, choose one, say $\alpha_t$, that is nonzero. Then, \begin{align*} \vect{u}_t &=\frac{-1}{\alpha_t}\left(-\alpha_t\vect{u}_t\right)&&\knowl{./knowls/property.MICN.knowl}{\text{Property MICN}}\\ &= \frac{-1}{\alpha_t}\left( \alpha_1\vect{u}_1+ \cdots+ \alpha_{t-1}\vect{u}_{t-1}+ \alpha_{t+1}\vect{u}_{t+1}+ \cdots+ \alpha_n\vect{u}_n \right)&&\knowl{./knowls/theorem.VSPCV.knowl}{\text{Theorem VSPCV}}\\ &= \frac{-\alpha_1}{\alpha_t}\vect{u}_1+ \cdots+ \frac{-\alpha_{t-1}}{\alpha_t}\vect{u}_{t-1}+ \frac{-\alpha_{t+1}}{\alpha_t}\vect{u}_{t+1}+ \cdots+ \frac{-\alpha_n}{\alpha_t}\vect{u}_n &&\knowl{./knowls/theorem.VSPCV.knowl}{\text{Theorem VSPCV}} \end{align*}

Since the values of $\frac{\alpha_i}{\alpha_t}$ are again scalars, we have expressed $\vect{u}_t$ as a linear combination of the other elements of $S$.

(⇐) Assume that the vector $\vect{u}_t$ is a linear combination of the other vectors in $S$. Write this linear combination, denoting the relevant scalars as $\beta_1$, $\beta_2$, …, $\beta_{t-1}$, $\beta_{t+1}$, …, $\beta_n$, as \begin{align*} \vect{u_t} &= \beta_1\vect{u}_1+ \beta_2\vect{u}_2+ \cdots+ \beta_{t-1}\vect{u}_{t-1}+ \beta_{t+1}\vect{u}_{t+1}+ \cdots+ \beta_n\vect{u}_n \end{align*}

Then we have \begin{align*} \beta_1\vect{u}_1 &+\cdots+ \beta_{t-1}\vect{u}_{t-1}+ (-1)\vect{u}_t+ \beta_{t+1}\vect{u}_{t+1}+ \cdots+ \beta_n\vect{u}_n\\ &=\vect{u}_t+(-1)\vect{u}_t&&\knowl{./knowls/theorem.VSPCV.knowl}{\text{Theorem VSPCV}}\\ &=\left(1+\left(-1\right)\right)\vect{u}_t&&\knowl{./knowls/property.DSAC.knowl}{\text{Property DSAC}}\\ &=0\vect{u}_t&&\knowl{./knowls/property.AICN.knowl}{\text{Property AICN}}\\ &=\zerovector&&\knowl{./knowls/definition.CVSM.knowl}{\text{Definition CVSM}} \end{align*}

So the scalars $\beta_1,\,\beta_2,\,\beta_3,\,\ldots,\,\beta_{t-1},\,\beta_t=-1,\beta_{t+1},\,\,\ldots,\,\beta_n$ provide a nontrivial linear combination of the vectors in $S$, thus establishing that $S$ is a linearly dependent set (Definition LICV).