Page 198 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Self Assessment

1. Let V be a finite dimensional vector space and W is any subspace of V. Prove that there is
1
a subspace W of V such that V = W  W .
2 1 2
2. True or false? If a diagonalizable operator has only the characteristic values 0 and 1, it is a
projection.
3. Let E , E , ... E be linear operators on the space V such that E + E + ... + E = I. Prove that
1 2 K 1 2 K
2
if E E = 0 for i  j, then E  E for each i.
i j i i
4. Let V be a finite dimensional vector space and let W ,... W be subspaces of V such that
1 K
V = W + W + ... + W and dim V = dim W + ... + W
1 2 K 1 K
Prove that V = W  W  ...  W .
1 2 K
16.3 Summary

 In this unit the importance is given to the ideas of invariant subspaces of a vector space V
for a linear operator T.
 The vector space V is decomposed into a set of linear invariant subspaces.

 The sum of the bases vectors of the invariant subspaces defines the basis vectors of the
vector space V.

16.4 Keywords

Skew-symmetric Matrices: Skew-symmetric matrices, i.e., matrices A such that A = –A.
t
Subspaces: These subspaces will be taken as independent subspaces of the vector space V and
after finding the independent basis of each independent subspace the ordered basis of the whole
space will be constructed.

16.5 Review Questions

1. If E , E are projections onto independent subspaces, then E + E is a projection. True or
1 2 1 2
false?
2
2
2. Let E , E be linear operators on the space V such that E + E = I, and E  E and E  E ,
1 2 1 2 1 1 2 2
then prove that E E = 0.
1 2
Answer: Self Assessment
2. Yes, true

16.6 Further Readings

Books Kenneth Hoffman and Ray Kunze, Linear Algebra
I.N. Herstein, Topics in Algebra

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