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Chapter 6
Linear Geometry

The reader can recall in Volume D the linear array and its algebraic properties were discussed, and then followed the application of arrays to geometry, producing position vectors. In previous chapters of the present volume the algebraic properties of matrices were discussed. In this chapter are discussions of geometric applications. Instead of horizontal linear arrays geometric points will be located by column arrays, that is:
    x        x
    y        y
              z
will locate points in the 2-dimensional coordinate plane and in the 3-dimensional coordinate space. It is convenient to write these collumn arrays as transposed linear row arrays:
            (x,y)'     and        (x,y,z)'
Here the apostrophe ' is used in place of  t.

Section 1:   Matrices as Functions

[1.1] (Linear expressions) A linear (homogenous) expression in x and y is an algebraic expression of the form   αx + βy,   where α and β are any real numbers. Similarly, a linear (homogeneous) expression in x,y, and z is an algebraic expression of the form   αx + βy + γz,   where α, β and γ are real numbers. The real numbers α, β and γ are called coefficients in those expressions.

The term "homogeneous" more accurately describes the linear expressions that will be discussed. But it will most often be omitted.

It is easy to extend this definition to include linear expressions in more variables, like x,y,z,w,... .However, this will not be done. Instead, discussions will involve only algebraic ideas that relate directly to the familiar geometric objects of point, line, plane and (3-dimensional) space.

In most discussions the coefficients of expressions will be integers, or, occasionally, rational numbers.

It is simple to find values for the linear expressions

5x + 6y    and    8x - 7y
if numerical values are given to x and y. If x and y are given values 3 and 4 respectively, then the expressions become
5(3) + 6(4) = 39    and    8(3) - 7(4) = - 4
By giving other values to x and y simultaneously, the two expressions have numerical values. Form the arrays (3,4)' and (39,-4)' . For the first array the linear expressions produce the second array. This action can be written
(3,4)' --> (39,-4)'
The reader can easily verify:
(1,-1)' --> (-1,15)',   (4,1)' --> (26,25)',   (α,β)' --> (5α + 6β, 8α - 7β)'
Consider these arrays as locating points in a plane. For each point (x,y)' in the plane the expressions assign a unique point (5x+6y,8x-7y)'. But this action satisfies the definition of a function F, carrying points in a plane onto points in the same plane:
F(x,y) = (5x+6y,8x-7y)
Here,
F(3,4) = (39,-4),   F(1,-1) = (-1,15),   F(4,1) = (26,25)
Using the coefficients of the linear expressions, form a 2x2 matrix:
            (matrix)    M =    5    6
                               8   -7
The following shows the details for the equation   M(x,y)'   =   (5x+6y,8x-7y)' :
                                           (5,6) * (x,y)'    =      5x + 6y
                                          (8,-7) * (x,y)'    =      8x - 7y
Intuitively speaking, M and F do the same thing to points in the plane. More exactly, M and F perform the same action on the coordinates of every point in the plane.


There is a similar discussion for 3x3 matrices and points in space. Given the linear expressions,
x + 2y + 3z,   4x + 5y + 6z,   7x + 8y + 9z
the function F can be formed: F(x,y,z) = (x+2y+3z,4x+5y+6z,7x+8y+9z) and the matrix is:
                                1    2    3
            (matrix)    M =     4    5    6
                                7    8    9
The reader can verify that the product   M(x,y,z)'   =   (x+2y+3z,4x+5y+6z,7x+8y+9z)'.


The linear expressions
2x + 3y    5x - 2y    -4x + y
receive values for x and y and return values for the three expressions. Therefore, they carry points from a plane into space:
(x,y)' --> (2x+3y,5x-2y,-4x+y)'
The function for this action is F defined by F(x,y) = (2x+3y,5x-2y,-4x+y).   The matrix is
                          2    3
            (matrix)  M = 5   -2
                         -4    1
For example,
M(6,7)' = (2(6)+3(7), 5(6)-2(7), -4(6)+1(7))' = (33, 14, -17)'
The same computation shows that   F(6,7) = (33, 14, -17).
By the matrix product a non-square matrix "changes" a column array of some length into a column array of a different length. The corresponding function carries an array of some length onto an array of a different length. Intuitively speaking, they both carry points in some dimension onto points in another dimension.

[1.2] (Corresponding functions) The correspoinding function F for a matrix M is obtained by defining F by F(x,y) = (linear array in x and y with coefficients from row 1 of M, linear array in x and y with coefficients from row 2 of M,....). If three variables are involved, F(x,y,z) = (linear array in x,y and z with coefficients from row 1 of M, linear array in x,y and z with coefficients from row 2 of M,....)



Section 2:   Linear Transformations

Subsets of geometric points can form recognizable objects: lines, circles, squares, cones,... . Some of these objects are more fundamental to discussions here: points, lines, planes, entire 3 dimensional space. They may be called basic affine objects. They have dimensions 0,1,2, and 3 respectively. They are fundamental parts of the subjects of plane and solid geometry. If they contain special points called origins, then they are basic linear objects. The term "basic" will often be omitted in these discussions.

Since linear objects have origins, position vectors exist. These have been denoted by p,q,r which locate points P,Q,R and can be expressed as   p = OP,   q = OQ,   r = OR.   For a geometric motivation for linear transformations click here.

[2.1] (Linear transformations) A function F from a linear object to a linear object is a linear transformation if and only if it satisfies the following two conditions:
  (a) F(p + q) = F(p) + F(q);     (homomorphism)
  (b) F(λp) = λF(p)     (homogeneous)
where p and q are any (position) vectors in the first linear object and λ is any real number.

Example:
Consider the function F from any linear object onto itself, defined by F(p) = 2p, for every point P in the first linear object.. Then F(q) = 2q, F(r) = 2r for position vectors q,r.... Intuitively speaking, F stretches any position vector to another vector pointing in the same direction but twice the length. F satisfies both conditions (a) and (b) of [2.1]:
  (a) F(p + q) = 2(p + q) = 2p + 2q = F(p) + F(q).
  (b) F(λp) = 2(λp) = λ(2p) = λF(p).

The following is a generalization of the defintion [2.1] of a linear transformation.

[2.2] Every linear transformation carries any linear combination of position vectors onto a linear combination orf position vectors.
Notation: If F is a linear transformation then F(λp + σq + ... + ωr) = λF(p) + σF(q) + ... + ωF(r).


In a linear object position vectors locate points, and to those points are "attached" linear arrays. Therefore, linear arrays can be represented by position vectors. If a position vector p in a plane locates a point P(x,y), then p = (x,y). Similarly, if p in space locates point P(x,y,z) then p = (x,y,z). As a result,definition [2.1] may be stated for linear arrays by replacing p and q by their equals (x1,y1) and (x2,y2):   F(p) = F(x1,y1), and F(q) = F(x2,y2). Similarly if F is from space: F(p) = F(x1,y1,z1), and F(q) = F(x2,y2,z2).   (Here the two F's denote different functions because they involve different linear objects, namely plane and space.)

Example: The function F defined by F(x,y) = (x + y, x - y) is a linear transformation from a plane onto itself. It carries point (1,1) onto point (2,0), point (5,3) onto (8,2). Click here to see the proof that it satisfies both conditions

(a)      F((x1,y1) + (x2,y2)) = F(x1,y1) + F(x2,y2)
and
(b)      F(λ(x,y)) = λF(x,y)                                  

The following functions F,G,H are also linear transformations:

F(x,y,z) = (2x + 3y + 4z, 4x - y - 3z, x +6y + z),      G(x,y,z) = (4x - y + z, 2x + 7y +5z),      H(x,y) = (x + y, x - y, 3x + 2y)
From the lengths of the arrays involved, it is easy to see that
F carries space into space,         G carries space into a plane;         H carries a plane into space

[2.3] A function from a plane into a linear object is a linear transformation if and only if carries an arbitrary point (x,y) onto a point whose coordinates are linear expressions of x and y. A function from space is a linear transformation if and only if it carries an arbitrary point (x,y,z) onto a point whose coordinates are linear expressions of x,y and z.



The pair of special unit arrays in a plane   (1,0) = i and (0,1) = j   as well as the triple of special unit arrays in space   (1,0,0) = i, (0,1,0) = j and (0,0,1) = k   play special roles with linear transformations.

Example: suppose F(1,0) = (4,5) and F(0,1) = (6,7). Onto what point does F carry the arbitrary point (x,y)? Notice that array (x,y) = x(1,0) + y(0,1). Then F(x,y) = xF(1,0) + yF(0,1) = x(4,5) + y(6,7) = (4x + 6y, 5x + 7y). Therefore, F carries the arbitrary point (x,y) onto the point (4x + 6y, 5x + 7y). This fact completely defines and determines F.

[2.4a] (Linear transformation from the plane is determined by images of two special points) From the identity

F(x,y) = xF(1,0) + yF(0,1)
a linear transformation F from the plane into a linear object is completely determined by the two image points F(1,0) and F(0,1).

[2.4b] (Linear transformation from space is determined by images of three special points) From the identity

F(x,y,z) = xF(1,0,0) + yF(0,1,0) + zF(0,0,1)
a linear transformation F from space into a linear object is completely determined by the three image points F(1,0,0), F(0,1,0), and F(0,0,1).

The expression xF(1,0) + yF(0,1) is actually an array whose coordinates are linear expressions in x and y. If F(1,0) = (α1, β1) and F(0,1) = (α2, β2) then

(#)        xF(1,0) + yF(0,1) = (α1x + β1y, α2x + β2y)
In a similar way, it is easy to show that if F(1,0,0) = (α1, β1, γ1), F(0,1,0) = (α2, β2, γ2) and F(0,0,1) = (α3, β3, γ3) then
(##)        xF(1,0,0) + yF(0,1,0) + zF(0,0,1) = (α1x + β1y + γz1, α2x + β2y + γz2, α3x + β3y + γ3z)
By eliminating or adding coordinates itis possible to produce further arguments supporting theorem [2.3].

The coefficients of the linear expressions in (#) and (##) can be collected into matrices M2 and M3:

[2.5a] (Linear transformations from a plane into itself and 2x2 matrices) If F is any linear transformation from a plane into a linear object then F(x,y)' = M(x,y)' where M is the matrix

	(matrix)   M =  coordinates of F(1,0)
			   coordinates of F(0,1)

[2.5b] (Linear transformations from space into itself and 3x3 matrices) If F is any linear transformation from space into a linear object then F(x,y,z)' = M(x,y,z)' where M is the matrix

	                   coordinates of F(1,0,0)
	(matrix)   M =     coordinates of F(0,1,0)
			      coordinates of F(0,0,1)

Suppose F is the linear transformation defined by F(x,y) = (3x + 4y, 5x + 6y). F(1,0) = (3(1) + 4(0), 5(1) + 6(0)) = (3,5), and F(0,1) = (4,6)



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In most situations the definition of a linear transformation requires coordinates. If F is defined as F(x,y) = (2x,3y) then F is a function from a plane onto itself). In words, for the array associated with a point, F doubles the first coordinate, whatever it is,   and   F triples the second coordinate, whatever it is.

To show F is a linear transformation, it is necessary to show conditions (a) and (b) of [2.1]. Let

p = (x1, y1)    and    q = (x2, y2)
Therefore,
p + q = (x1+x2,   y1+y2)   and   λp = (λx, λy)

For condition (a)

F(p + q) = F((x1,y1) + (x2,y2)) = F(x1+x2, y1+y2) =   ...   = F(x1,y1) + F(x2,y2) = F(p) + F(q)
For the complete details of this chain of equalities click here.

For condition (b)

F(λp) = F(λx,λy) = (2λx,3λy) = λ(2x,3y) = λF(x,y) = λF(p)



Another example is: F(x,y) = (2x + 3y, 3x - y). Since arrays of length two are involved, this function carries a line into (actually onto) a line. It can be proven directly, but theorem [2.3] below covers this situation. The theorem also provides the proof that F(x,y,z) = (3x + 2y + z, x -2y + 3z, 2x - y - 4z) defines a linear transformation. The following is a generalization of conditions (a) and (b) of defintion [2.1]. It is also a step toward establishing a connection between linear transformations and matrices.

[2.2] A linear transformation carries lincombs onto lincombs.

Let F be a linear transformation, and let   αp + βq + γr + ...   be any lincomb of vectors p, q, r, ... . Then

F(αp + βq + γr + ...) = F(αp) + F(βq) + F(γr) + ... = αF(p) + βF(q) + γF(r) + ...
This last sum is a lincomb of vectors F(p), F(q), F(r),... .


Recall the idea of linear expressions:
     αx + βy is a linear expression in x and y
     αx + βy + γz is a linear expression in x,y and z.

[2.3] If the coordinates of the image points of a function F(x,y) are linear expressions of x and y, then F is a linear transformation from a plane into (possibly onto) a linear object. Similarly, if the coordinates of the image points of a function F(x,y,z) are linear expressions of x,y,z, then F is a linear transformation from space into (possibly onto) a linear object.
Notation:
if F(x,y) = αx
   then F is a linear transformation from a plane to a point (origin)

If F(x,y) = (α1x + β1y,   α2x + β2y)
    then F is a linear transformation from a plane to itself, a line in that plane or a point (origin)

if F(x,y,z) = (α1x + β1y + γ1z,   α2x + β2y + γ2z,   α3x + β3y + γ3z)
    then F is a linear transformation from space to itself, to a plane in that space, to a line in that space or to a point (origin).


Recall that the position vectors

i = (1,0),   j = (0,1) form a basis for the plane and
i = (1,0,0),   j = (0,1,0),   k = (0,0,1) form a basis for space
This means that any position vector (array) can be written as a lincomb of these basic vectors (arrays):
(x,y) = x(1,0) + y(0,1) and
(x,y,z) = x(1,0,0) + y(0,1,0) + z(0,0,1)
Then for any linear transformation F,
F(x,y) = xF(1,0) + yF(0,1) and
F(x,y,z) = xF(1,0,0) + yF(0,1,0) + zF(0,0,1)
Now these images
F(1,0),   F(0,1)   and
F(1,0,0),   F(0,1,0),   F(0,0,1)
are fixed points somewhere in some linear object. These points have attached arrays that can form row arrays in a matrix:
	(matrix)			M =  F(1,0)
		                         F(0,1)     and
 

	(matrix) 				F(1,0,0)
					M = 	F(0,1,0)
						F(0,0,1)
Then
			F(x,y) = M(x,y)'     and
			F(x,y,z) = M(x,y,z)'