192 CHAPTER 9. LINEAR TRANSFORMATIONS

9.2.3 ProjectionsIn Physics it is important to consider the work done by a force field on an object. Thisinvolves the concept of projection onto a vector. Suppose you want to find the projection ofa vector v onto the given vector u, denoted by proju (v) This is done using the dot productas follows.

proju (v) =(v ·u

u ·u

)u

Because of properties of the dot product, the map v7→proju (v) is linear,

proju (αv+βw) =

(αv+βw ·u

u ·u

)u = α

(v ·uu ·u

)u+β

(w ·uu ·u

)u

= α proju (v)+β proju (w) .

Example 9.2.5 Let the projection map be defined above and let u =(1,2,3)T . Does thislinear 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 thevector in Rn which has a 1 in the ith position and a zero everywhere else. Thus a typicalvector x =(x1, · · · ,xn)

T can be written in a unique way as

x =n

∑j=1

x je j.

From the way you multiply a matrix by a vector, it follows that proju (ei) gives the ith

column of the desired matrix. Therefore, it is only necessary to find

proju (ei)≡( ei·u

u ·u

)u

For the given vector in the example, this implies the columns of the desired matrix are

114

 123

 ,214

 123

 ,3

14

 123

 .

Hence the matrix is

114

 1 2 32 4 63 6 9

 .

9.2.4 Matrices Which Are One To One Or OntoLemma 9.2.6 Let A be an m×n matrix. Then A(Fn) = span(a1, · · · ,an) where a1, · · · ,andenote the columns of A. In fact, for x = (x1, · · · ,xn)

T ,

Ax =n

∑k=1

xkak.