Reading a 3D transformation

A 3 by 3 matrix sends a vector mathbfvmathbf{v} to a new point mathbfv′=Amathbfvmathbf{v'} = Amathbf{v} by combining the three columns of the matrix.

Each column shows where a basis direction lands, so a matrix can rotate, scale, reflect, or shear in three dimensions.

The length comparison is useful when you want to know whether the matrix preserves magnitude.

Where this helps

  • Testing rotation-like matrices in graphics and robotics.
  • Checking whether a matrix stretches a direction or keeps its length.
  • Understanding how column vectors encode transformed axes.

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