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Suppose $v_1, v_2, v_3$ are (row) vectors in $\mathbb{R}^3$, and they are not parallel, then what you can say about the rank of the matrix:

\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}

The answer says $\text{rank } A > 1$, where $A$ is the matrix.

But why is this true?

From the definition of rank

Rank: The maximum number of linearly independent row vectors in the matrix. Both definitions are equivalent.

Clearly, if $v_1, v_2, v_3$ are not parallel, then the set $\{v_1, v_2, v_3\}$ is a linearly independent set, so $\text{rank }A = 3$ should hold shouldn't it?

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  • $\begingroup$ Do you consider $(1,0,0)$, $(0,1,0)$, $(1,1,0)$ parallel? $\endgroup$ – Hagen von Eitzen Feb 5 '17 at 23:26
  • $\begingroup$ @HagenvonEitzen, how? They arent scalar multiples of one another? I would consider ex $(1, 0, 0), (,2, 0, 0), (3, 0, 0)$ parallel $\endgroup$ – Amad27 Feb 5 '17 at 23:29
  • $\begingroup$ @Amad27 Exactly, they aren't parallel. They also aren't linearly independent. $\endgroup$ – Leon Sot Feb 6 '17 at 0:45

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