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This seems easy, but I can't find a standard name. Suppose I have a vector space defined a set M of linearly independent columns and N rows (let's assume that M >= N and the columns are linearly independent.)

I want to split this vector space into two subspaces based on a set of k < M "constraint" directions C1...Ck, and the remaining M-k directions Rk_1....m. (For describing the kinematics of constrained motion.)

I think the set Ck is usually called the "nullspace" or "kernel" of the subspace. Question is: what is the remaining subspace, spanned by Rk, (whose vectors are all orthogonal to each of the Cks) subspace called?

I've seen "range" and "image" used for related ideas, but I'd like to know if there is a standard term.

Arse Grammatica
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  • Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – José Carlos Santos Sep 10 '21 at 07:36
  • If you want to be precise in terminology, please take a bit more care with things discussed in the question. A vector space is an abstract notion; rows and columns have no place there (these belong to matrices). No vector space is "defined [by] a set of columns" either (even less by a set of columns AND a set of rows; do you mean by a given matrix?). Are you talking about the span of the columns in $\Bbb R^N$? Also, if a matrix has more columns than rows, the columns cannot be linearly independent. Even reading generously, your question is not at all clear; what vector space is split and how? – Marc van Leeuwen Sep 10 '21 at 07:56
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    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Sep 10 '21 at 08:00
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    I am not quite sure what your nullspace (kernel) is the nullspace of, there usually needs to be a matrix $A$ for that, but $\text{Ker}(A)^\perp=\text{Ran}(A^*)$. So your complement is the range of its Hermitian adjoint (transpose, if the entries are real). – Conifold Sep 10 '21 at 08:14
  • @MarcvanLeeuwen Actually the terminology used by OP is something that you would learn in linear algebra class for engineers. Namely definitions of a vector space using a set of columns and somewhat mixing columns/rows of matrices and vectors are quite common. I agree that the question is a bit confusing, maybe an example would help. – Korf Sep 10 '21 at 10:30

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