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I was just thinking about this question

Given a $2 \times 3$ matrix

$$\begin{bmatrix}{1}&{3}\\{3}&{4}\\{5}&{6}\end{bmatrix}$$

Would this be considered $3$ vectors in $2$ dimensional space or $2$ vectors in $3$ dimensional space? If so why is the column space a plane?

This is confusing me. Pardon my fundamentally phrased question if it is. I am new to linear algebra and this is not getting around my head.

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It can be considered as either of those things depending on whether you are considering the column space (which would be a $2$ dimensional plane living in $\mathbb{R}^3$ because we have two linearly independent vectors, the span of which forms a $2$-dimensional linear subspace (i.e. a plane)) or the row space (which consists of $3$ vectors in $\mathbb{R}^2$ only two of which are linearly independent, so they span the whole space).

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    So there is no right or wrong way to answering this question? – 2017-01-19
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    Yes, there is no right or wrong answer. – 2017-01-19
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    I have given the answer to "why is the column space a plane?", and the answer to your first question is that it can be considered either depending on whether you are considering the row or column space. – 2017-01-19
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    @AndrewWhelan I am referring to the first question. Yes what I am looking for is that it can be perceived either way with no hard and fast rule like what John said – 2017-01-19