Linear Algebra, Theory And Applications, 2012a

236 LINEAR TRANSFORMATIONS
Let T be the name of this linear transformation. In this case, T e 3 = e 3 ,Te 1 =
(cos θ, sin θ, 0) T , and T e 2 =(− sin θ, cos θ, 0) T . Therefore, the matrix of this transformation
is just
⎛
⎞
cos θ − sin θ 0
⎝ sin θ cos θ 0 ⎠ (9.5)
0 0 1
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 proj u (v) Thisisdoneusingthedot
product as follows.
( v · u
)
proj u (v) = u
u · u
Because of properties of the dot product, the map v → proj u (v) is linear,
proj u (αv+βw) =
( )
αv+βw · u
u = α
u · u
= α proj u (v)+β proj u (w) .
( v · u
)
u + β
u · u
( w · u
)
u
u · u
Example 9.3.22 Let the projection map be defined above and let u =(1, 2, 3) T . Find the
matrix of this linear transformation with respect to the usual basis.
You can find this matrix in the same way as in earlier examples. proj u (e i )givesthei th
column of the desired matrix. Therefore, it is only necessary to find
( ei·u
)
proj u (e i ) ≡ u
u · u
For the given vector in the example, this implies the columns of the desired matrix are
1
14
⎛
⎝
1
2
3
⎞ ⎛
⎠ , 2 ⎝
14
1
2
3
⎞ ⎛
⎠ , 3 ⎝
14
1
2
3
⎞
⎠ .
Hence the matrix is
⎛
1
⎝
14
1 2 3
2 4 6
3 6 9
⎞
⎠ .
Example 9.3.23 Find the matrix of the linear transformation which reflects all vectors in
R 3 through the xz plane.
As illustrated above, you just need to find T e i where T is the name of the transformation.
But T e 1 = e 1 ,Te 3 = e 3 , and T e 2 = −e 2 so the matrix is
⎛
⎝ 1 0 0 −1 0
0
⎞
⎠ .
0 0 1
Example 9.3.24 Find the matrix of the linear transformation which first rotates counter
clockwise about the positive z axis and then reflects through the xz plane.