Page 196 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes If any (and hence all) of the conditions of the last lemma hold, we say that the sum W = W + ...
1
+ W is direct or that W is the direct sum of W ,..., W and we write
k 1 k
W = W    W
1 k
In the literature, the reader may find this direct sum referred to as an independent sum or the
interior direct sum of W ,..., W .
1 k

Example 1: Let V be a finite-dimensional vector space over the field F and let { ,...,  }
1 n
be any basis for V. If W is the one-dimensional subspace spanned by , then V = W    W .
i i 1 n
Example 2: Let n be a positive integer and F a subfield of the complex numbers, and let
V be the space of all n × n matrices over F. Let W be the subspace of all symmetric matrices, i.e.,
1
t
matrices A such that A = A. Let W be the subspace of all skew-symmetric matrices, i.e., matrices
2
t
A such that A = –A. Then V = W  W . If A is any matrix in V, the unique expression for A as a
1 2
sum of matrices, one in W and the other in W , is
1 2
A = A + A
1 2
1
A = (A  A t )
1
2
1
A = ( – A t )
A
2
2
Example 3: Let T be any linear operator on a finite-dimensional space V. Let c ,.., c be the
1 k
distinct characteristic values of T, and let W be the space of characteristic vectors associated with
i
the characteristic value c . Then W ,..., W are independent. In particular, if T is diagonalizable,
i 1 k
then V = W    W .
1 k
2
Definition: If V is a vector space, a projection of V is a linear operator E on V such that E = E.
Suppose that E is a projection. Let R be the range of E and let N be the null space of E.
2
1. The vector  is in the range R if and only if E = . If  = E, then E = E  = E = .
Conversely, if  = E, then (of course)  is in the range of E.
2. V = R  N.
3. The unique expression for  as a sum of vectors in R and N is  = E + ( – E).
From (1), (2), (3) it is easy to see the following. If R and N are subspaces of V such that V = R  N,
there is one and only one projection operator E which has range R and null space N. That
operator is called the projection on R along N.
Any projection E is (trivially) diagonalizable. If { ,...,  } is a basis for R and { ,...,  } a basis
1 r r+1 n
for N, then the basis  = { ,...,  } diagonalizes E.
1 n
I  0
E
[ ] 
  
 0 0 
where I is the r × r identity matrix. That should help explain some of the terminology connected
3
with projections. The reader should look at various cases in the plane R (or 3-space, R ), to
2
convince himself that the projection on R along N sends each vector into R by projecting it
parallel to N.

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