# How to know if $\mathrm{Im} A = \mathbb R^n$ given a matrix?

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?

• 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. Mar 28, 2019 at 0:54

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."

• "Though it is four dimensional, it is not the whole of "four-dimensional space"" -- I think this is misleading... Mar 28, 2019 at 1:31
• @amsmath How so? Is there a way that this could be better expressed? Mar 28, 2019 at 1:35
• "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) Mar 28, 2019 at 1:38
• @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! Mar 28, 2019 at 1:49
• The thing is that you can very easily confuse people that are not really safe with maths because they are often insecured. Mar 28, 2019 at 1:55