Princeton University MATH 202

FEB 13, 2008

• Lots of definitions this week.

• The length of a vector v is given by

the square root of its dot product against itself. An unit vector is a vector with length 1. For
an arbitrary non-zero vector v, its corresponding unit vector is given by .

• The dot product between two non -zero vectors measures how “parallel” they are. In particular,
if the dot product between u and v is 0, then they are perpendicular.

• Remember that a linear transformation corresponds, geometrically, to transformations that
sends lines to lines and the origin to the origin. If we look at the linear transformation from
the plane R² to itself, there are essentially five types, as illustrated be low .1

• A scaling transformation is given by

T(x) = kx

for some positive number k. Its transformation matrix is given by

In the illust ration , k = 2.

• Projection: given a line L (through the origin) in the plane, we ask “how much of a given
vector v is parallel to the line L?” Projection unto a line L is to “squash away” all components
that are perpendicular to L, and keeping the comp onent that is parallel to L. Let u be
an unit vector of L, the formula for the projection is

T(x) = (x · u)u

If (since |u| = 1, a² + b² = 1), the corresponding matrix is

• ( Draw pictures of projections of figures)

• Using the projection, given a unit vector u, we can write any vector v as the sum of two
parts : the part that is parallel to u and the part that is perpendicular. The parallel part is from
the projection: (v · u)u. The perpendicular part we’ll just write as v − (v · u)u. You can
easily check that

[v − (v · u)u] · u = 0

and

[v − (v · u)u] + (v · u)u = v

Figure 1: Examples of planar linear transformations: (a) is the original image; (b) is a scaling by a
factor of 2; (c) is a reflection about a horizontal line; (d) is the projection unto a vertical axis; (e) is
a shear of the original, where the point (x, y) is send to (x + y, y); and (f) is a counter-clockwise
rotation of the original by 56°.

Reflection: suppose we have a line L in the plane, and we want to reflect our figure over that
line L. How do we do it?

– First let’s look at the case when L is the horizontal axis. To reflect over the horizontal,
or x-, axis, we send the y coordinate of a figure to −y. So the transformation looks like

With the associated matrix looking like

– For reflection over the line L (which has its unit vector u), we first decompose a vector
v into the parts parallel to and perpendicular to u like above . To reflect over L, we
keep the part that is parallel the same, and negate the part that is perpendicular to u. So
we have the transformation is given by

T(x) = (x · u)u + (−1)[x − (x · u)u] = 2(x · u)u − x

– Again, supposing the unit vector , the associated matrix looks like

• Shear: a shear is achieved by slicing up the plane into parallel lines, and then rearranging the
lines by shifting each successive line a little bit more.

– A vertical shift is given by the matrix

– A horizontal shift is given by

– A commonly seen example is the Galilean transform. A point in space-time is transformed
by

This transformation relates vectors as seen in by two different observers , one moving
at a velocity v relative to the other.

– In general, one recognizes a shear transformation when the transformation fixes some
direction (T(w) = w), and that an arbitrary vector is shifted by a multiple of that
direction (T(x) − x is in the direction of w).

– (This bullet will not be on the test. It is not to be taught in class. Included for the
reader’s benefit.) Given unit vectors u,w such that they are perpendicular u · w = 0.
And given a non-zero constant k, the shear transform that fixes the w direction is given
by

T(x) = x + k(x · u)w

If and (up to multiplying w by (−1), this is the canonical form
of two unit vectors in a plane that are perpendicular.), then the associated matrix is

• Rotation: the matrix for rotation of an angle θ in the counterclockwise direction is

Notice that A has the form

with a² + b² = 1. Any matrix that has this form is a matrix of a rotation.

• By applying various of the above transformations successively, we can get more complicated
linear transformations. Notice: the order matters ! For example, consider a counterclockwise
rotation by 90° and a vertical shear. Starting with the unit vector in the horizontal direction, if
we first rotate then shear, the result is the unit vector in the vertical direction (since a vertical
shear does not change a vector in the vertical direction). But if we first shear then rotate,
we’ll get a vector sitting in the second quadrant.

• Generalizations to higher dimensions:

– The scaling transformation is just a constant times the identity transformation, both of
which is available in higher dimensions, so extending it to higher dimensions is easy.

– Similarly, the projection expression

T(x) = (x · u)u

for some unit vector u is also available in higher dimensions, but the corresponding
matrix becomes more complicated.

– The reflection transformation is slightly tricky. The idea is that once you are given a
vector and a line to reflect the vector over, there are two cases: the vector is parallel to
the line or not. In the first case, since the vector is parallel to the line, its reflection is
itself, so nothing it to be done. If the vector is not on the line, then the vector and the
line together defines at least three non-colinear points, which then define a plane. The
reflection of the vector is defined as the reflection in that plane over the line. While the
geometrical interpretation is slightly more complicated, the algebraic expression is still
the same:

T(x) = 2(x · u)u − x

and, as for the projection, the associated matrix is complicated.

– The shear and rotation transformations are much more complicated. The general algebraic
expression given for the shear (which is not covered in class) above in terms
of two perpendicular unit vectors can be used to generalize the shear transformation to
higher dimensions. The rotation is more complicated because one also need to specify
which axis to rotate around. (But illustrate with rotation about a fixed coordinate axis,
which is simple .)