In learning about the general formula for a circle: $$(x-h)^2 + (y-k)^2 = r^2$$ my book states that $3$ points are sufficient to guarantee the solution (or absence of solution) for an unique circle.
It follows this up by explaining that $3$ different points creates $3$ different equations...namely:
1) $(x_1-h)^2 + (y_1-k)^2 = r^2\;$ where $\;(x_1,y_1)\;$ is the first point provided
2) $(x_2-h)^2 + (y_2-k)^2 = r^2\;$ where $\;(x_2,y_2)\;$ is the second point provided
3) $(x_3-h)^2 + (y_3-k)^2 = r^2\;$ where $\;(x_3,y_3)\;$ is the third point provided
I'm okay with this as I'm familiar with the whole "$3$ equations for $3$ unknowns".
However, the book's explanation concludes with the statement:
"The equations can't be linearly dependent since the points that lead to them are all different". I am not particularly familiar with linear algebra so I do not understand the significance of this statement. Is there a visual representation of this statement that can help me understand what they mean?