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So I know that the set of vectors $Ax = 0$ is called the null space or the kernel of A; and I know that we call the set of vectors b such that $Ax = b$ does have a solution the range or the column space of A.

But, what do we call the set of vectors b such that $Ax = b$ has no solution? Do we have such a name? I feel like there's a connection between the set of vectors b that have no x solution and the null space.

FafaDog
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    Whatever the name, keep in mind that such a set is not a vector space: It doesn't contain $0$. – Ningxin May 22 '20 at 04:30
  • Ah, interesting. Both the null space and the range/column space of A both contain the zero vector. Is it safe to say that the column space and the null space are not disjoint? – FafaDog May 22 '20 at 04:36
  • If b is the zero vector, then the column space of A should contain the null space. Is that correct? – FafaDog May 22 '20 at 04:36
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    The column space and null space of $A$ only depends on $A$. If $b$ is non-zero, then the set of vectors $x$ such that $Ax=b$ is not a vector space. Instead, it is an affine space. – Ningxin May 22 '20 at 04:38
  • Oh! Very interesting! Does that mean the column space of a matrix isn't a vector space as well? – FafaDog May 22 '20 at 04:48
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    The column space, by definition, is the vector space spanned by the columns. – Ningxin May 22 '20 at 04:56
  • So if the column space is a vector space, then it must contain the zero vector. Same for the null space. But the column space should contain the null space correct? Because those would be the set of vectors such that Ax = b =0. – FafaDog May 22 '20 at 05:34
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    Don’t limit your thinking to square matrices and endomorphisms. The null space is a subspace of the domain; the column space a subspace of the codomain. In general those aren’t the same vector space, so there’s no overlap even possible between the column and null spaces. When the domain and codomain are the same, the only thing that you can say for sure is the the zero vector is an element of both. – amd May 22 '20 at 05:37

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Assuming you have $T_A:X\to Y$ via $T_A\mathbf{x} = A\mathbf{x}$, the set $Y\setminus \operatorname{im} T_A$ is what you describe. But it doesn't have a name that I'm aware of. It isn't a vector space (because it doesn't contain $\mathbf 0$). And it totally depends on what you regard as the larger space $Y$ containing $\operatorname{im} T_A$ is, so it isn't natural (not being determined by $A$) and probably not very interesting either.

MPW
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