Before looking a some specific examples of types of linear operators, a slight variation on the idea. Linear operators can be assumed to operate on points in 2-space or 3-space rather than on vectors. The connection? the vectors are just taken to be the position vectors of the points. So for example, the linear operator

T(xi + yj) = (x + y)i + 2xj

can also be written for points as

T(x, y) = (x + y, 2x).

You can then look at how the transformation changes a collection of points, say a picture, rather than a collection of vectors. Each approach is useful and equivalent to the other.

In the diagrams on this page, we illustrate the effect of linear operators on both points and vectors by showing both the point and its position vector, like this:

Since rotations are rigid, they rotate the diagonal of a parallelogram along with its sides, so sums are preserved.

In 3-space, rotations about any line through the origin are also linear operators, again because they just transform vectors rigidly.
Rotations    A rotation about the origin in 2-space is a linear operator.
Also, a vector and its scalar multiple are just rotated to a different orientation, so scalar multiples are preserved.

 

 

Reflections    A reflection in a line through the origin in 2-space is a linear operator. Reflections are rigid, like rotations; they just change all vectors into their mirror images in the line, so sums reflect into sums and scalar multiples reflect into scalar multiples.

In 3-space, there are several types of reflection: you can reflect

  • in a plane
  • in a line
  • through the origin.

All of these reflections are linear operators.

You can also reflect through the origin in 2-space. That replaces every vector by its negative, so reflection through the origin is also a linear operator.

 

Scale changes   In either 2-space of 3-space, suppose you multiply every vector by the same positive scalar. This transformation just changes the size of any vector configuration - parallelograms just turn into bigger or smaller parallelograms, etc. and a vector and its scalar multiple just turn into a different sized vector and its scalar multiple.

So scale changes are linear operators. Click the button to apply a scale change to the vectors.

A scale change by a factor c > 0 has the formula T(u) = cu for all vectors u. It's easy to check algebraically that T is a linear operator from this equation.

 

 

Projections

Recall what a projection is: the "shadow' of one vector in the direction of another. Specifically, the projection of the vector v onto the line through the origin with direction vector u is given by the formula

.

(See the learning object on Dot Products for more details.)

Projections onto a line through the origin in 2-space are also linear operators, but it's difficult to see this from a picture, since all "sum parallelograms" get flattened onto the line. However, we can easily prove that projections are linear operators algebraically.

Suppose we're projecting onto a line with a normalized direction vector a (i.e. a has length 1, so a•a = ||a||2 = 1). Then the formula for the projection is

T(u) = proja(u) = (a•u)a.

Then for any two vectors u and v in 2-space

T(u + v) = proja(u + v)
            = (a•[u + v])a
            = (a•u + a•v)a
            = (a•u)a + (a•v)a
            = proja(u) + proja(v)
            = T(u) + T(v).

Also, for any vector u in 2-space and any scalar c,

T(cu) = proja(cu)
        = (a•[cu])a
        = c(a•u)a
        = c proja(u)
        = cT(u).

So projections preserve both sums and scalar multiples, and are thus linear operators.

In 3-space, there are two types of projections: projections onto lines through the origin and projections onto planes through the origin. Both are linear operators.

There are two "trivial" linear operators in 2-space and in 3-space: the zero operator and the identity operator.

The zero operator O takes every vector into the zero vector: O(u) = 0 for all vectors u. It is a linear operator because

  • it preserves sums: for any vectors u and v,
             O(u) + O(v) = 0 + 0 = 0 = O(u + v).
  • it preserves scalar multiples: for any scalar c and any vector u,
             cO(u) = c0 = 0 = 0 = O(cu).

The identity operator I transforms every vector into itself: I(u) = u for all vectors u. It's clear geometrically that I must be a linear operator, and algebraically,

  • it preserves sums: for any vectors u and v,
             I(u + v) = u + v = I(u) + I(v) .
  • it preserves scalar multiples: for any vector u and any scalar c,
             I(cu) = cu = cI(u).

Neither of these operators "does" much, but they become useful when discussing the algebra of linear operators, much like the numbers 0 and 1 do in arithmetic.

Notice that the identity operator can be considered to be a rotation (through angle 0) or a scale change (with scale factor 1). It is not a projection or a reflection.