by bob jantzen
Our textbook Edwards, Penney and Calvis, Differential Equations and Linear Algebra (fourth edition), avoids talking about linear changes of coordinates on the Rn vector spaces which are the arena on which systems of linear differential equations play out as well as where we can associate mental pictures with solving the initial value problem for a single higher order linear differential equation, the latter of which may be interpreted as a linear change of coordinates on the space of initial data. These linear changes of coordinates are at the heart of the eigenvector decoupling mechanism and allow eigendirections to be visualized in the direction fields associated with systems of linear differential equations. Two dimensional examples are enough to give these direct visualizations of what we do when we apply the "recipe" to solve these DE systems.
A problem that both students and faculty face is that all of our mathematical foundations rely on the Euclidean geometry of the Rn spaces, from geometry and trigonometry through all levels of calculus. In our MAT2705 course, we subtract away geometry from the linear mathematics to rely only on the basic linear operations of vector addition and scalar multiplication combined together into linear combination. Lengths and angles no longer are relevant and we are naturally led to employ coordinate systems with nonorthogonal axes and arbitrary (but linear) tickmarks along those axes disconnected from lengths. These coordinate systems allow us to directly convert the matrix and Rn vector space manipulations of chapters 3 and 4 into tangible representations that in the 2-dimensional case is associated with graph paper. The rectangular grids of Cartesian coordinate systems are directly related to the vector space properties (i.e., linearity properties) of their associated natural bases. Allowing more general linear changes of coordinates leads to the corresponding grids based on parallelograms and parallelopipeds which can help in more general linear problems in the same way the rectangular coordinates and grids help us in calculus.
It is enough to work in the plane R2 with the natural basis e1 = <1,0> = i and e2 = <0,1> = j, with associated Cartesian coordinates (x1,x2) that we usually call (x,y). Any vector in the plane can be represented as a linear combination of the basis vectors with coefficients which we call coordinates with respect to the basis (when we think of the elements of R2 as "points" unrelated to any choice of origin) or components of the vector with respect to the basis (when we think of the elements of R2 as "vectors", namely as directed line segments from the origin to the point in question, "arrows"). I will use the term coordinates and components interchangeably, but reserve angle bracket delimiters to the vector interpretation (points in R2 cannot be added with explicitly using its vector space structure, which means a choice of origin to nail down the space and establish position vectors which locate the points).
Thus x = <x1,x2>= x1 e1 + x2 e2 . We can repeat this decomposition of a vector using any other basis of the plane, namely any other two linearly independent vectors b1 = <b11,b21>, b2 = <b12,b22> and a new coordinate vector which follows from the coefficients of the expansion: x = y1 b1 + y2 b2 = y1 <b11,b21> + y2 <b12,b22> .
By agreeing to think of vectors as column matrices when we incorporate them into matrix relationships, we get a basis changing matrix B = <b1 | b2 > whose columns in order are the new ordered basis vectors, and the two distinct "expansions" of the generic vector x lead to x = B y since right multiplication of a matrix by a coefficient vector takes a linear combination of the columns of that matrix. This gives us the coordinate transformation from known new coordinates to the corresponding old coordinates of any point in the plane, but typically we start out knowing the old coordinates of points in our natural coordinate system on the space so we need to invert this to go the opposite direction: y = B-1 x .
Rather than thinking of particular numbers in the specification of a particular vector in the plane x = <x1,x2>= x1 e1 + x2 e2 , we can think of the coordinates x1,x2 as the Cartesian coordinate functions on the plane which pick out the first and second components of any particular vector in the plane (technically these linear functions are elements of the dual vector space). We then have a new pair of coordinate functions y1,y2 on the plane associated with the new basis and these truly become relationships between old and new coordinate functions in the plane: a linear change of coordinates on the plane.
The natural way of visualizing the Cartesian coordinate functions is through their associated coordinate grids where we choose an interval along each axis to mark off equal tickmarks and we then (in the two dimensional case) consider the lines of constant values of the coordinate marked off along each axis, which are parallel to the other axis. This leads to a 2-parameter family of intersecting lines which in the Cartesian case are orthogonal in the natural Euclidean geometry that we introduce to go beyond linearity but which we do not want to use now. Since our original standard basis vectors are unit vectors in this geometry, the tickmarks mark off arclength along their corresponding axes. For example, using unit intervals for the tickmarks we get an integer grid consisting of grid lines whose intersection points (the "grid points", which is the set of all integer coefficient linear combinations of the basis vectors) all have integer coordinates. We can also use other powers of 10 for different length scales (say refining the grid by dividing unit intervals into tenths), but let's stay simple. The old coordinate axes are simply the directed lines through the origin containing the respective basis vectors, namely all real multiples of the corresponding basis vector. Since they have a positive direction, it is natural to put an arrow head only at the end of the coordinate axis in a given plot window corresponding to that positive direction (horizontal increases to the right, vertical increases upwards).
We can repeat this for the new basis except that we choose a particular multiple of our basis vectors to mark off the tickmarks, for example integer multiples of the basis vectors along each axis. The new coordinate axes are directed lines through the origin containing their respective basis vector, and positive multiples pick out the positive direction, so one can indicate the positive direction on each axis by an arrowhead at that end of the axis in a plot window. Students in high school seem to have been taught universally to put arrowheads on both ends of the axes to indicate that they continue out to infinity perhaps, but this habit must be broken now that we need to know which is the positive direction when our axes can point in any direction in the plane.
Once you have this grid for the new coordinate system, if you pick any grid point on the new grid (where the new coordinates will be integers), then following the two gridlines back to the new coordinate axes allows one to read off how many multiples of each basis vector go into the "projection" of the original vector onto those axes by a parallelogram construction---these are the new coordinates. Taking any point in the plane, drop parallels to each of the coordinate axes to the other coordinate axis and you will obtain the two vector projections whose vector sum (main diagonal of the parallelogram they form as two edges coming out of the origin) is the original vector being projected. No geometry, only linear combination goes into the projection process. First vector projection to establish the two vectors parallel to the basis vectors which add up to the original vector, then scalar multiplication of the basis vectors to reach those two projected vectors along the basis vectors. The scalar multiples are the new coordinates.
So many words, no pictures! We need to look at the visual representations of these words.
more examples showing new coordinate grids: