Page 254 - 35Linear Algebra

254 Orthonormal Bases and Complements

• The standard basis vectors are orthogonal (in other words, at right
angles or perpendicular);
T
e i e j = e e j = 0 when i 6= j
i
This is summarized by

1 i = j
T
e e j = δ ij = ,
i
0 i 6= j
where δ ij is the Kronecker delta. Notice that the Kronecker delta gives the
entries of the identity matrix.
Given column vectors v and w, we have seen that the dot product v w is
T
n
the same as the matrix multiplication v w. This is an inner product on R .
T
We can also form the outer product vw , which gives a square matrix. The
outer product on the standard basis vectors is interesting. Set
Π 1 = e 1 e T
1
 
1
0
 

=  . 1 0 · · · 0
 
.
 .
0
 
1 0 · · · 0
 0 0 · · · 0 

=  . . . .

 .
.
0 0 · · · 0
. . .
Π n = e n e T
n
 
0
0
 
=  . 0 0 · · · 1
 
.
 .
1
 
0 0 · · · 0
 0 0 · · · 0 
 
=  . .
 . . . .
0 0 · · · 1

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