We want to prove that any set of \(t+1\) or more vectors from \(V\) is linearly dependent. So we will begin with a totally arbitrary set of vectors from \(V\text{,}\) \(R=\set{\vectorlist{u}{m}}\text{,}\) where \(m\gt t\text{.}\) We will now construct a nontrivial relation of linear dependence on \(R\text{.}\)

Each vector \(\vectorlist{u}{m}\) can be written as a linear combination of the vectors \(\vectorlist{v}{t}\) since \(S\) is a spanning set of \(V\text{.}\) This means there exist scalars \(a_{ij}\text{,}\) \(1\leq i\leq t\text{,}\) \(1\leq j\leq m\text{,}\) so that
\begin{align*}
\vect{u}_1&=a_{11}\vect{v}_1+a_{21}\vect{v}_2+a_{31}\vect{v}_3+\cdots+a_{t1}\vect{v}_t\\
\vect{u}_2&=a_{12}\vect{v}_1+a_{22}\vect{v}_2+a_{32}\vect{v}_3+\cdots+a_{t2}\vect{v}_t\\
\vect{u}_3&=a_{13}\vect{v}_1+a_{23}\vect{v}_2+a_{33}\vect{v}_3+\cdots+a_{t3}\vect{v}_t\\
&\quad\quad\vdots\\
\vect{u}_m&=a_{1m}\vect{v}_1+a_{2m}\vect{v}_2+a_{3m}\vect{v}_3+\cdots+a_{tm}\vect{v}_t
\end{align*}

Now we form, unmotivated, the homogeneous system of \(t\) equations in the \(m\) variables, \(x_1,\,x_2,\,x_3,\,\ldots,\,x_m\text{,}\) where the coefficients are the just-discovered scalars \(a_{ij}\text{,}\)
\begin{align*}
a_{11}x_1+a_{12}x_2+a_{13}x_3+\cdots+a_{1m}x_m&=0\\
a_{21}x_1+a_{22}x_2+a_{23}x_3+\cdots+a_{2m}x_m&=0\\
a_{31}x_1+a_{32}x_2+a_{33}x_3+\cdots+a_{3m}x_m&=0\\
\vdots\quad\quad&\\
a_{t1}x_1+a_{t2}x_2+a_{t3}x_3+\cdots+a_{tm}x_m&=0
\end{align*}

This is a homogeneous system with more variables than equations (our hypothesis is expressed as \(m\gt t\)), so by Theorem HMVEI there are infinitely many solutions. Choose a nontrivial solution and denote it by \(x_1=c_1,\,x_2=c_2,\,x_3=c_3,\,\ldots,\,x_m=c_m\text{.}\) As a solution to the homogeneous system, we then have
\begin{align*}
a_{11}c_1+a_{12}c_2+a_{13}c_3+\cdots+a_{1m}c_m&=0\\
a_{21}c_1+a_{22}c_2+a_{23}c_3+\cdots+a_{2m}c_m&=0\\
a_{31}c_1+a_{32}c_2+a_{33}c_3+\cdots+a_{3m}c_m&=0\\
\vdots\quad\quad&\\
a_{t1}c_1+a_{t2}c_2+a_{t3}c_3+\cdots+a_{tm}c_m&=0
\end{align*}

As a collection of nontrivial scalars, \(c_1,\,c_2,\,c_3,\,\dots,\,c_m\) will provide the nontrivial relation of linear dependence we desire,
\begin{align*}
&\lincombo{c}{u}{m}\\
&=c_{1}\left(a_{11}\vect{v}_1+a_{21}\vect{v}_2+a_{31}\vect{v}_3+\cdots+a_{t1}\vect{v}_t\right)&&
\knowl{./knowl/definition-SSVS.html}{\text{Definition SSVS}}\\
&\quad\quad+c_{2}\left(a_{12}\vect{v}_1+a_{22}\vect{v}_2+a_{32}\vect{v}_3+\cdots+a_{t2}\vect{v}_t\right)\\
&\quad\quad+c_{3}\left(a_{13}\vect{v}_1+a_{23}\vect{v}_2+a_{33}\vect{v}_3+\cdots+a_{t3}\vect{v}_t\right)\\
&\quad\quad\quad\quad\vdots\\
&\quad\quad+c_{m}\left(a_{1m}\vect{v}_1+a_{2m}\vect{v}_2+a_{3m}\vect{v}_3+\cdots+a_{tm}\vect{v}_t\right)\\
&=c_{1}a_{11}\vect{v}_1+c_{1}a_{21}\vect{v}_2+c_{1}a_{31}\vect{v}_3+\cdots+c_{1}a_{t1}\vect{v}_t&&
\knowl{./knowl/property-DVA.html}{\text{Property DVA}}\\
&\quad\quad+c_{2}a_{12}\vect{v}_1+c_{2}a_{22}\vect{v}_2+c_{2}a_{32}\vect{v}_3+\cdots+c_{2}a_{t2}\vect{v}_t\\
&\quad\quad+c_{3}a_{13}\vect{v}_1+c_{3}a_{23}\vect{v}_2+c_{3}a_{33}\vect{v}_3+\cdots+c_{3}a_{t3}\vect{v}_t\\
&\quad\quad\quad\quad\vdots\\
&\quad\quad+c_{m}a_{1m}\vect{v}_1+c_{m}a_{2m}\vect{v}_2+c_{m}a_{3m}\vect{v}_3+\cdots+c_{m}a_{tm}\vect{v}_t\\
&=\left(c_{1}a_{11}+c_{2}a_{12}+c_{3}a_{13}+\cdots+c_{m}a_{1m}\right)\vect{v}_1&&
\knowl{./knowl/property-DSA.html}{\text{Property DSA}}\\
&\quad\quad+\left(c_{1}a_{21}+c_{2}a_{22}+c_{3}a_{23}+\cdots+c_{m}a_{2m}\right)\vect{v}_2\\
&\quad\quad+\left(c_{1}a_{31}+c_{2}a_{32}+c_{3}a_{33}+\cdots+c_{m}a_{3m}\right)\vect{v}_3\\
&\quad\quad\quad\quad\vdots\\
&\quad\quad+\left(c_{1}a_{t1}+c_{2}a_{t2}+c_{3}a_{t3}+\cdots+c_{m}a_{tm}\right)\vect{v}_t\\
&=\left(a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3}+\cdots+a_{1m}c_{m}\right)\vect{v}_1&&
\knowl{./knowl/property-CMCN.html}{\text{Property CMCN}}\\
&\quad\quad+\left(a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3}+\cdots+a_{2m}c_{m}\right)\vect{v}_2\\
&\quad\quad+\left(a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}+\cdots+a_{3m}c_{m}\right)\vect{v}_3\\
&\quad\quad\quad\quad\vdots\\
&\quad\quad+\left(a_{t1}c_{1}+a_{t2}c_{2}+a_{t3}c_{3}+\cdots+a_{tm}c_{m}\right)\vect{v}_t\\
&=0\vect{v}_1+0\vect{v}_2+0\vect{v}_3+\cdots+0\vect{v}_t&&
c_j\text{ as solution}\\
&=\zerovector+\zerovector+\zerovector+\cdots+\zerovector&&
\knowl{./knowl/theorem-ZSSM.html}{\text{Theorem ZSSM}}\\
&=\zerovector&&
\knowl{./knowl/property-Z.html}{\text{Property Z}}\text{.}
\end{align*}

That does it. \(R\) has been undeniably shown to be a linearly dependent set.

The proof just given has some monstrous expressions in it, mostly owing to the double subscripts present. Now is a great opportunity to show the value of a more compact notation. We will rewrite the key steps of the previous proof using summation notation, resulting in a more economical presentation, and even greater insight into the key aspects of the proof. So here is an alternate proof — study it carefully.

Alternate Proof: We want to prove that any set of \(t+1\) or more vectors from \(V\) is linearly dependent. So we will begin with a totally arbitrary set of vectors from \(V\text{,}\) \(R=\setparts{\vect{u}_j}{1\leq j\leq m}\text{,}\) where \(m\gt t\text{.}\) We will now construct a nontrivial relation of linear dependence on \(R\text{.}\)

Each vector \(\vect{u_j}\text{,}\) \(1\leq j\leq m\) can be written as a linear combination of \(\vect{v}_i\text{,}\) \(1\leq i\leq t\) since \(S\) is a spanning set of \(V\text{.}\) This means there are scalars \(a_{ij}\text{,}\) \(1\leq i\leq t\text{,}\) \(1\leq j\leq m\text{,}\) so that
\begin{align*}
\vect{u}_j&=\sum_{i=1}^{t}a_{ij}\vect{v}_i&&1\leq j\leq m
\end{align*}

Now we form, unmotivated, the homogeneous system of \(t\) equations in the \(m\) variables, \(x_j\text{,}\) \(1\leq j\leq m\text{,}\) where the coefficients are the just-discovered scalars \(a_{ij}\text{,}\)
\begin{align*}
\sum_{j=1}^{m}a_{ij}x_j=0&&1\leq i\leq t
\end{align*}

This is a homogeneous system with more variables than equations (our hypothesis is expressed as \(m\gt t\)), so by Theorem HMVEI there are infinitely many solutions. Choose one of these solutions that is not trivial and denote it by \(x_j=c_j\text{,}\) \(1\leq j\leq m\text{.}\) As a solution to the homogeneous system, we then have \(\sum_{j=1}^{m}a_{ij}c_{j}=0\) for \(1\leq i\leq t\text{.}\) As a collection of nontrivial scalars, \(c_j\text{,}\) \(1\leq j\leq m\text{,}\) will provide the nontrivial relation of linear dependence we desire,
\begin{align*}
\sum_{j=1}^{m}c_{j}\vect{u}_j
&=\sum_{j=1}^{m}c_{j}\left(\sum_{i=1}^{t}a_{ij}\vect{v}_i\right)&&
\knowl{./knowl/definition-SSVS.html}{\text{Definition SSVS}}\\
&=\sum_{j=1}^{m}\sum_{i=1}^{t}c_{j}a_{ij}\vect{v}_i&&
\knowl{./knowl/property-DVA.html}{\text{Property DVA}}\\
&=\sum_{i=1}^{t}\sum_{j=1}^{m}c_{j}a_{ij}\vect{v}_i&&
\knowl{./knowl/property-C.html}{\text{Property C}}\\
&=\sum_{i=1}^{t}\sum_{j=1}^{m}a_{ij}c_{j}\vect{v}_i&&
\knowl{./knowl/property-CMCN.html}{\text{Property CMCN}}\\
&=\sum_{i=1}^{t}\left(\sum_{j=1}^{m}a_{ij}c_{j}\right)\vect{v}_i&&
\knowl{./knowl/property-DSA.html}{\text{Property DSA}}\\
&=\sum_{i=1}^{t}0\vect{v}_i&&
c_j\text{ as solution}\\
&=\sum_{i=1}^{t}\zerovector&&
\knowl{./knowl/theorem-ZSSM.html}{\text{Theorem ZSSM}}\\
&=\zerovector&&
\knowl{./knowl/property-Z.html}{\text{Property Z}}\text{.}
\end{align*}

That does it. \(R\) has been undeniably shown to be a linearly dependent set.