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I recently read this question regarding matrix-multiplication, and the answer mentions that :

... is not a "change in reference system" (because lots of nonzero vectors go to zero, but there is no way to just "change your perspective" and start seeing a nonzero vector as zero) but is a linear transformation nonetheless.

1) I am not able to understand what does change in reference system mean. What all counts as a change in reference system?

2) Why some linear transformations aren't a change in reference system?

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nishantbhardwaj2002
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2 Answers2

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It may help with the intuition if we make this a little more concrete. Let's just consider what happens in $\mathbb R^3,$ treated as Cartesian coordinates of three-dimensional space.

A change in reference system would be a different way of measuring the coordinates to all points in space. This happens for example in a first-person video game where you're flying an aircraft or spaceship; as you roll, pitch, and yaw your craft the game has to change the view on the screen accordingly. It does this by taking the coordinates of each object in the "world" frame and figuring out their new coordinates in your changed reference system: how far to the right, "above", or in front of you each object should appear. If you put your craft at the origin of coordinates and remain there while you rotate, the change in coordinates is a linear transformation (the zero vector remains zero). Then instead of locating each point in space by taking a linear combination of the three unit vectors along the $x,$ $y,$ and $z$ axes, the game figures out a linear combination of the three unit vectors "forward," "right," and "up" (from your rotated orientation) to reach the same point.

There can also be less "natural" changes in reference frame in which we measure the distance along each axis differently, or in which the three axes are not perpendicular to each other. But we're still able to travel some number of units in the direction of the first axis, then some number of units in the direction of the second axis, then some in the direction of the third axis, to get to any point in space that we want.

A common characteristic of all such changes of reference frame is that we can change them back simply by performing another linear transformation on the coordinates. That is, every point in space has coordinates in the new reference frame, and there is a way to recover the original coordinates of that point from these new coordinates by multiplying them by a matrix.

But the kinds of linear transformations the quote is about, which take more than one vector to zero, are sometimes called projections, because they map all of space onto a single plane or a single line. The transformation by a matrix of all zeros, which maps everything to the origin, is also such a transformation. When we apply a projection, we cannot use the transformed axes to describe all points in space, and we cannot "undo" the transformation just by applying another linear transformation.

David K
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  • Awesome answer!!! (Q1) change in reference systems where zero vector doesn't remain zero (flying aeroplane, where we have also lifted up from the origin) aren't linear transformations (we can't have a matrix corresponding to that transformation, where a zero vector becomes non zero)? (Q2) a reference system got by merging z-axis into y-axis, would it be a valid example of linear transformation corresponding to projections? here we can't revert back to the 3-d point, as it becomes sort of a 2-d point in new reference system – nishantbhardwaj2002 Mar 12 '17 at 08:07
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    (1) When the origin also moves, you can have an affine transformation. There are various ways of representing that, but they generally come down to some way of combining linear transformations with translations (where you simply add the same vector to every set of coordinates). – David K Mar 12 '17 at 14:59
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    (2) Yes, merging the z-axis onto the y-axis via a linear transformation is an example of a projection. Everything ends up in the x-y plane, and if all you know is the image of a point, you have no way to know exactly where it came from. – David K Mar 12 '17 at 15:00
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Reference frames can be thought of as follws:

There are geometrical objects, e.g. a vector $x$ and an observer $A$ who sees these objects. In his reference frame this observer assign coordinates $x^\mu$ to the vectors.

Another observer $B$ who possibly moves or is located at another position sees the same vector $x$, but assigns possibly different coordinates $x^\nu$.

These coordinates are related by a linear map, i.e. a matrix $M$ with entries $M^\mu_\nu$, such that $x^\mu=\sum_\nu M^\mu_\nu x^\nu$. However, from the physical situation, not all linear maps correspond to a valid change of reference frames.

For example, if a vector in one coordinate frame is zero in all coordinates $x^\mu$, this can be thought of that it does not exist, but then it cannot be observed in the other reference frame i.e. all its coordinates must $x^\nu$ must be zero too. This condition is fulfilled, if $\det M\ne 0.$

Above conditions needs to be always fulfilled, but depending if one considers for example Newtonian mechanics or special relativity, the valid changes are either Galilean transformations or Poincaré transformations.

klirk
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