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I understand that 2D matrix representing the lines in 2D space gives a unique solution where those lines intersect. Same in 3d, unique solution is where the planes intersect.

Can someone explain what is happening to those lines or planes when we do one of the row operations of adding a scalar multiple of one row to another row?

I understand that

  1. scalar multiplying a row still maintains the same line. Also,

  2. switching rows keeps the same lines, but I don't understand the

  3. effect of adding a scalar multiple of one row to another row - what happens here geometrically?? How to understand it? Why can we make this operation and still have the same solution at the end?

Thank you.

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You can interpret a linear sistem as a set of “conditions“. When you add a scalar multiple (Say 3) of a row to a pre-existent row you’re saying: points that are vanishing in this row must vanish also in 3 times the other row. This process keeps the solutions stable. In other words, the adding process gives an extra (And) condition to the Row: points have to do that AND that. But the second “that” is redundant, because is a condition already considered.

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