[ 6 marks] For the following linear operators, flind one invariamt subspace of the givea dimescion. (a) For
D:P_(3)(R)->P_(3)(R)
, the differentiation operator (so
D(p(x))=p^(')(x)
), find an invariant subspace of dimension 2 . (b) For
S:C^(2)->C^(2)
give by
S(v)=[[1,2],[-1,3]]v
, find an invariant subspace of dimension 1 . (c) For
T
:
M_(2\times 2)(R)=>M_(2\times 2)(R)
, given by
T(A)F^(T)A^(T)
End an invariant subepace of dimension 3 .

Question

[ 6 marks] For the following linear operators, flind one invariamt subspace of the givea dimescion. (a) For

D:P_(3)(R)->P_(3)(R)

, the differentiation operator (so

D(p(x))=p^(')(x)

), find an invariant subspace of dimension 2 . (b) For

S:C^(2)->C^(2)

give by

S(v)=[[1,2],[-1,3]]v

, find an invariant subspace of dimension 1 . (c) For

T

:

M_(2\times 2)(R)=>M_(2\times 2)(R)

, given by

T(A)F^(T)A^(T)

End an invariant subepace of dimension 3 .

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