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Ways of viewing Y = AX

The linear system

or, more concisely,

Y = AX

can be viewed in different ways. One may have seen trick drawings in which one first sees the head of a monkey and then on a second look he sees an elephant. Y = AX is similar. It can be viewed in different ways. Sometimes one thinks of it one way, sometimes another. Some of the ways it can be viewed are:

1. As a change in variables from the variables y1, y2, ..., yn of the vector Y to the variables x1, x2, ..., xn of the vector X.

2. As a point transformation that maps points (and figures) from one space into another space (or the same space). It effects a linear transformation on the points in the space and when viewed in this way as a point transformation we call it a “linear point transformation” to emphasize how we are viewing it.

3. As a change to an oblique coordinate system which has a different metric than the usual rectangular Cartesian system — with the unit of length varying from one axis to another. Changing notation from Y = AX to X = A X' with X' referring to the new oblique system and X referring to the original system the linear transformation X = AX' can be viewed as a change to an oblique system in which the elementary unit vectors of the new oblique system correspond to the column vectors of matrix A. Stated in different words, we change to a different basis, the new basis corresponding to an oblique coordinate system. Thus instead of thinking of the figure as changing as we do in View 2, we think of the figure as remaining as it is and the coordinate system in which the figure is expressed as changing.
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