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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.

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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 .

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