Page 255 - 35Linear Algebra

14.2 Orthogonal and Orthonormal Bases 255

In short, Π i is the diagonal square matrix with a 1 in the ith diagonal position
1
and zeros everywhere else .
T
T
T
Notice that Π i Π j = e i e e j e = e i δ ij e . Then:
j
j
i

i = j
Π i
Π i Π j = .
0 i 6= j
Moreover, for a diagonal matrix D with diagonal entries λ 1 , . . . , λ n , we can
write
D = λ 1 Π 1 + · · · + λ n Π n .

14.2 Orthogonal and Orthonormal Bases

There are many other bases that behave in the same way as the standard
basis. As such, we will study:

• Orthogonal bases {v 1 , . . . , v n }:

v i v j = 0 if i 6= j .

In other words, all vectors in the basis are perpendicular.

• Orthonormal bases {u 1 , . . . , u n }:

u i u j = δ ij .

In addition to being orthogonal, each vector has unit length.

n
Suppose T = {u 1 , . . . , u n } is an orthonormal basis for R . Because T is
a basis, we can write any vector v uniquely as a linear combination of the
vectors in T;
n
1
v = c u 1 + · · · c u n .
Since T is orthonormal, there is a very easy way to find the coefficients of this
linear combination. By taking the dot product of v with any of the vectors

1
This is reminiscent of an older notation, where vectors are written in juxtaposition.
This is called a “dyadic tensor”, and is still used in some applications.

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