Canonical Forms Linear Algebra Notes

Note Ei are well defined projections with
range(Ei) = Wi and NEi = Wi
where Wi = W1 ⊕ · · · ⊕ Wi−1 ⊕ Wi+1 ⊕ · · · ⊕ Wk.
Following is a theorem on projections.
6.5 (Theorem) Let V be a finite dimensional vector space over a
field F. Suppose V = W1 ⊕ W2 ⊕ · · · ⊕ Wk be direct sum of subspaces
Wi. Then there are k linear operators E1, . . . , Ek on V such that
1. each Ei is a projection (i. e. E 2 i = Ei).
2. EiEj = 0 ∀ i = j.
3. E1 + E2 + · · · + Ek = I.
4. range(Ei) = Wi.
Conversely, if E1, . . . , Ek are k linear operators on V satisfying all
the conditions (2)-(3) above then Ei is a projection (i.e. (1) holds)
and with Wi = Ei(V ) we have V = W1 ⊕ W2 ⊕ · · · ⊕ Wk.
Proof. The proof is easy and left as an exercise. First, try with
k = 2 operators, if you like.
Homework: page 213, Exercise 1, 3, 4-7, 9.
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