Linear Algebra

Section VI. Projection 251
�v
c�p�s
�v − c�p�s
We can solve for this coefficient by noting that because �v − c�p�s is orthogonal to
a scalar multiple of �s it must be orthogonal to �s itself, and then the consequent
fact that the dot product (�v − c�p�s) �s is zero gives that c�p = �v �s/�s �s.
1.1 Definition The orthogonal projection of �v into the line spanned by a
nonzero �s is this vector.
�v �s
proj [�s ](�v) = · �s
�s �s
Exercise 19 checks that the outcome of the calculation depends only on the line
and not on which vector �s happens to be used to describe that line.
1.2 Remark The wording of that definition says ‘spanned by �s ’ instead the
more formal ‘the span of the set {�s }’. This casual first phrase is common.
1.3 Example In R2 , to orthogonally project into the line y =2x, wefirstpick
a direction vector for this line. For instance,
� �
1
�s =
2
will do. With that, calculation of a projection is routine.
� � � �
� �
2 1
2
� �
�v =
3
3 2
� � � �
1
· =
1 1 2
2 2
8
5 ·
� � � �
1 8/5
=
2 16/5
1.4 Example In R3 , the orthogonal projection of a general vector
⎛ ⎞
x
⎝y⎠
z
into the y-axis is
⎛
⎝ x
⎞
y⎠
z
⎛
⎝ 0
⎞
1⎠
0
which matches our intuitive expectation.
⎛
⎝ 0
⎞
1⎠
0
⎛
⎝ 0
⎛
⎞ · ⎝
1⎠
0
0
⎞ ⎛
1⎠
= ⎝
0
0
⎞
y⎠
0