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If W is a subspace of , then the transformation that maps each vector x in into its orthogonal
projection in W is called the orthogonal projection of on W.
We leave it as an exercise to show that orthogonal projections are linear operators. It follows from Formula 5 that the
standard matrix for the orthogonal projection of on W is
(6)
where A is constructed using any basis for W as its column vectors.
EXAMPLE 3 Verifying Formula (6)
In Table 5 of Section 4.2 we showed that the standard matrix for the orthogonal projection of on the -plane is
(7)
To see that this is consistent with Formula 6, take the unit vectors along the positive x and y axes as a basis for the -plane,
so that
We leave it for the reader to verify that is the identity matrix; thus Formula 6 simplifies to
which agrees with 7.
EXAMPLE 4 Standard Matrix for an Orthogonal Projection
Find the standard matrix for the orthogonal projection P of on the line l that passes through the origin and makes an angle
with the positive x-axis.
Solution as a basis for this
The line l is a one-dimensional subspace of . As illustrated in Figure 6.4.3, we can take
subspace, so
We leave it for the reader to show that is the identity matrix; thus Formula 6 simplifies to
