T-admissible- If T is a linear operator on a vector space V and W is a subspace of V then W is T-admissible if
(i) W is invariant under T.
(ii) If f(T) β is in W, there exists γ ∈W :
f(T) β = F(T)γ.
Jordan matrix (Jordan form)- A matrix having the elements of its principal diagonal equal and not zero, the elements immediately above (or below) those in the diagonal unity and all other elements zero.
Decompositions:
Independent subspaces- If W1 … Wk are k subspaces of vector space V. Then W1 … Wk are independent if α1 +… + αk = 0, αi∈ Wi implies each αi = 0.
Projection- If V is a vector space, a projection of V is a linear operator E on V such that
E2 = E.
Nilpotent- If N is a linear operator on the vector space V. Then N is nilpotent if for some positive integer r, Nr = 0.
Direct sum- If W1 … Wn are independent subspaces of vector space V. The sum
W = W1 + W2 +… + Wn is called direct and W is the direct sum of W1 … Wn denoted by
W = W1 ⊕ W2 ⊕… ⊕ Wn
(i) W is invariant under T.
(ii) If f(T) β is in W, there exists γ ∈W :
f(T) β = F(T)γ.
Jordan matrix (Jordan form)- A matrix having the elements of its principal diagonal equal and not zero, the elements immediately above (or below) those in the diagonal unity and all other elements zero.
Decompositions:
Independent subspaces- If W1 … Wk are k subspaces of vector space V. Then W1 … Wk are independent if α1 +… + αk = 0, αi∈ Wi implies each αi = 0.
Projection- If V is a vector space, a projection of V is a linear operator E on V such that
E2 = E.
Nilpotent- If N is a linear operator on the vector space V. Then N is nilpotent if for some positive integer r, Nr = 0.
Direct sum- If W1 … Wn are independent subspaces of vector space V. The sum
W = W1 + W2 +… + Wn is called direct and W is the direct sum of W1 … Wn denoted by
W = W1 ⊕ W2 ⊕… ⊕ Wn
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