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I have recently started learning linear algebra. I have come across the concepts of vector spaces and their bases. One type of problem I am now encountering is finding the basis and dimension of solution space of a given system of linear equations.

The confusion I've is why we would be able to consider the solution space a vector space. Because without it being a vector space it ceases to have a basis, right? Or am I missing something plainly obvious?

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Yes, it has to be a vector space in order to have a basis.

If your linear system of equations is homogeneous, meaning that the right-hand-side is the $0$-vector, then the solution set will always be a subspace of whatever space it resides in.

Why is this true? Any linear system of equations can be written in terms of a matrix:

$$Ax = 0.$$

The solution set consists of all vectors $x$ that solves this equation, so all you need to check is that the subspace axioms hold.

Certainly $A0 = 0$. If $Ax_1=0$ and $Ax_2=0$, then because $A$ works linearly, we get $A(x_1+x_2) = Ax_1+Ax_2 = 0+0=0$. And finally, if $c$ is a scalar, then again by linearity, we get $A(cx) = cAx = c0 = 0$.

In conclusion, the solution set of $Ax=0$ is indeed a subspace. If $A$ is $m\times n$, then it will be a subspace of $\mathbb{R}^n$.

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