Let α={e1=(1,0,0),e2=(0,1,0),e3=(0,0,1)} be the canonical basis of R 3 and consider the linear operator T:R 3 →R 3 that satisfies T(e1)=3e 1 +e 3 T(e2)=2e 2 T(e3)=2e 2 −e 3 It is correct to say that:...

Let α={e1=(1,0,0),e2=(0,1,0),e3=(0,0,1)} be the canonical basis of R³
and consider the linear operator T:R³→R³
that satisfies

T(e1)=3e₁+e₃

T(e2)=2e₂

T(e3)=2e₂−e₃

It is correct to say that:

a) The characteristic polynomial of T is p(λ)=−(2−λ)(3−λ)(1+λ) and the eigenspaces V−1, V2 and V3 of T satisfy dim(V−1)+dim(V1)+dim(V3)=3.

b) The eigenvalues ​​of T are −1, 2 and 3 and the eigenspaces V−1, V2 and V3 of T satisfy dim(V−1)⋅dim(V1)⋅dim(V3)=3.

c) The eigenvalues ​​of the operator T are −1, 2 and 3 and the eigenspaces V−1, V2 and V3 of T satisfy dim(V−1)+dim(V1)+dim(V3)=2.

d) The characteristic polynomial of T is p(λ)=−(2−λ)(3−λ)(1+λ) and the eigenspaces V−1, V2 and V3 of T satisfy dim(V−1)+dim(V1)+dim(V3)=2.

e) The eigenvalues ​​of operator T are 0, 2 and 3 and the eigenspaces V0, V2 and V3 of T satisfy dim(V0)+dim(V1)+dim(V3)=3.