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Suppose I do the following to a vector in $\mathbb{R}^{2}$

  1. Reflect across x-axis
  2. Reflect across y-axis
  3. Reflect across origin

Now to do the tranformations one by one we could left-multiply the vector by the corresponding tranformation matrix to get final tranformed vector having final tranformation as the effect the three.

If I have to do the transformation in one step then I would need a tranformation matrix representing the combined effect of all the three transformation. Let us define the tranformation as A, B and C for relecting across x-axis, y-axis and the origin respectively.

As far as I know we have to take product of three matrices as (C.B.A) to get the combined effect.

My question is why are we multiplying it rather than adding it? What geometric interpretations it would have if we added the matrix instead of multiplying as I mentioned?

akshay
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1 Answers1

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That's because the matrix of the composition is the product of the matrices. The sum of two matrices corresponds to the sum of two linear maps, not to their composition. Besides, if it did, then the composition $f\circ g$ of two linear maps $f$ and $g$ would always be equal to $g\circ f$, but, in general, it is not.

José Carlos Santos
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  • Could you please tell geometrically what would happen if we added the matrices? You said "sum of two matrices corresponds to the sum of two linear maps", I am finding it difficult ot visualize geometrically. – akshay Apr 03 '21 at 16:06
  • The sum of two linear maps $f,g\colon\Bbb R^2\longrightarrow\Bbb R^2$ is the map$$\begin{array}{rccc}f+g\colon&\Bbb R^2&\longrightarrow&\Bbb R^2\&v&\mapsto&f(v)+g(v).\end{array}$$So we are adding the vectors the vectors $f(v)$ and $g(v)$, which, geometrically, can be expressed through the parallelogram law of vector addition. – José Carlos Santos Apr 03 '21 at 16:13