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[Community Question] Linear-algebra: Vector Space as the set of solutions of matrix equation AX=O

One of our user asked:

One of my professor's lecture notes on Vector Spaces start by the following lines:-

We have seen that if $det(A)$ = 0, then system $AX=O$ has infinite number of solutions. We shall now see that in this case, the set of solutions has a structure called vector space.

My doubt is in what sense do the set of an infinite number of the solution of equation $AX = O$ (given |A|=0) is actually a structure of Vector Space? How does the term Vector Space come into picture?


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