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Knowing that I have $5\times 6$ matrix $A$ with $\dim \mathrm{Im}\, A = 4$ and $\dim \ker A= 2$, I am asked if $\mathrm{Im} \,A = \mathbb R^4$? The answer is that $\mathrm{Im}\, A$ is a subspace of $\mathbb R^5$. Is it because we have 5 rows? If not why?

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    $\begingroup$ Yes, we have 5 rows and six columns. Therefore, $A$ as a linear mapping maps from $\mathbb R^6$ to $\mathbb R^5$. Hence, $Im(A)$ is a subspace of $\mathbb R^5$. The space $\mathbb R^4$ is another object and not a subspace of $\mathbb R^5$. It can be embedded though, but the result will not be $\mathbb R^4$ anymore. $\endgroup$
    – amsmath
    Mar 28, 2019 at 0:54

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Any $m\times n$ matrix can be thought of as describing a linear transformation from $\mathbb R^n$ to $\mathbb R^m$. The fact that $A$ is a $5 \times 6$ matrix tells us that its column space (or, image) will be a subspace of $\mathbb R^5$. Since $\dim \mathrm{Im}\, A = 4$, we know that this subspace has dimension four, but this is not the same as $\mathbb R^4$.

Consider the the following subspace of $\mathbb R^3$:

$$ V=\mathrm{span}\, \left\{ \begin{bmatrix}1\\1\\0\end{bmatrix}, \begin{bmatrix}0\\1\\1\end{bmatrix}\right\} $$

It is easy to see that the subspace $V$ is a plane situated in $\mathbb R^3$. However, this plane is clearly not the same as $\mathbb R^2$; it is a separate object altogether. The same is true for the column space of your matrix. Though it has four dimensions, it is not the "four-dimensional space," but rather a subsection of "five-dimensional space."

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  • $\begingroup$ "Though it is four dimensional, it is not the whole of "four-dimensional space"" -- I think this is misleading... $\endgroup$
    – amsmath
    Mar 28, 2019 at 1:31
  • $\begingroup$ @amsmath How so? Is there a way that this could be better expressed? $\endgroup$ Mar 28, 2019 at 1:35
  • $\begingroup$ "Not a whole of" somehow indicates that it is a part of four-dimensional space, which is of course not true. Rather the contrary. It is a "whole" four-dimensional space. But just not $\mathbb R^4$. ;o) $\endgroup$
    – amsmath
    Mar 28, 2019 at 1:38
  • $\begingroup$ @amsmath I see your point, though I feel that my phrasing adequately conveys that sentiment, especially with the surrounding context. Nonetheless, I have edited my post to remove any possible ambiguity for future readers. Thanks for your close review! $\endgroup$ Mar 28, 2019 at 1:49
  • $\begingroup$ The thing is that you can very easily confuse people that are not really safe with maths because they are often insecured. $\endgroup$
    – amsmath
    Mar 28, 2019 at 1:55

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