Relation between the row picture and column picture

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I am going through Strang's book "Linear Algebra and its applications" and it's written there that, " If n planes have no point in common or have infinitely many points, then the columns lie in the same plane".
This is a very important idea, one that unifies the row and the column picture. Can someone explain to me why this happens? Why is it that when the row picture fails, the column picture fails inevitably?