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There is this line in book "Mathematics for Machine Learning":

" Every subspace $U ⊆ (R^n , +, ·)$ is the solution space of a homogeneous system of linear equations Ax = 0 for x ∈ $R^n$ "

I read this post. Every subspace of $\mathbb{R}^n$ is a solution space of a homogeneous system of linear equation.

I still haven't understood the statement. I understand the solution of this is null space. How can every possible subspace be the solution?

I am getting the feeling I either didn't understand the English or am lacking some serious foundation in linear algebra. I just started studying linear algebra.

Kailash Yadav
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    because of the existence of quotient space, see https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra) – Q-Zhang Jun 23 '22 at 07:47

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Of course every subspace is not the solution set of the same homogeneous system of linear equations; I think that is how you may have been reading this sentence. Similarly, if one says: "every subspace has a basis" that does not mean they all have the same basis (nor by the way that each of them has only one basis). So the proper way to read the sentence is: for every subspace $U$ of $\Bbb R^n$, there exists a (at least one) matrix $A$ such that $U=\{\, x\in\Bbb R^n\mid Ax=0\,\}$.

Marc van Leeuwen
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