Linear independence of vectors and solutions to systems of equations

Question

I am wondering whether the following argument is correct: "Let $V$ be a set of vectors in $\mathbb{R}^m$, $$ V=\{v_1,v_2,…, v_n\} $$ Then if $n>m$, the vectors in $V$ are linearly dependent and $$ λ_1v_1+λ_2v_2+ \ldots + λ_nv_n=0 $$ has only the trivial solution." Shouldn't the second part state that the equation has infinitely many solutions including the trivial one? Thank you very much in advance for your answers.

Answer 1

It should say "...has non-trivial solution", because $\;\{v_1,...,v_n\}\;$ being linearly dependent means exactly that, namely: there exist scalars $\;a_i\in\Bbb R\;$ not all zero s.t.

$$\sum_{k=1}^n a_kv_k=0$$

Since the field ($\;\Bbb R\;$) is infinite, this in fact means there are infinite non-trivial choices of the above scalars .