Question

If 0 is an eigenvalue of a matrix a then the set of columns of a is linearly independent?

Answer 1

A⋅v⃗ =λ⋅v⃗ 0⃗ =λ⋅v⃗ −A⋅v⃗ 0⃗ =(λ−A)⋅v⃗ 0⃗ =(λ⋅I−A)⋅v⃗ A⋅v→=λ⋅v→0→=λ⋅v→−A⋅v→0→=(λ−A)⋅v→0→=(λ⋅I−A)⋅v→

and the determinant of (λ⋅I−A)(λ⋅I−A) must be 00, or in other words (λ⋅I−A)(λ⋅I−A) is not invertible, or in other words the columns of (λ⋅I−A)(λ⋅I−A) are linearly dependent, or the nullspace of (λ⋅I−A)(λ⋅I−A) is non trivial.

Could someone explain me better these statements? What's the relation between a statement and the other?

I understood some stuff, but some other clarifications might help too.

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