In Physics it is important to consider the work done by a force field on an object. This involves the concept
of projection onto a vector. Suppose you want to find the projection of a vector, v onto the given vector, u,
denoted by Pu
This is done using the dot product as follows.
Because of properties of the dot product, the map v → Pu
Example 5.3.4 Let the projection map be defined above and let u =
T. Does this linear
transformation come from multiplication by a matrix? If so, what is the matrix?
You can find this matrix in the same way as in the previous example. Let ei denote the vector in ℝn
which has a 1 in the ith position and a zero everywhere else. Thus a typical vector, x =
be written in a unique way as
From the way you multiply a matrix by a vector, it follows that Pu
column of the desired
matrix. Therefore, it is only necessary to find
For the given vector in the example, this implies the columns of the desired matrix are
Hence the matrix is