- #1
noospace
- 75
- 0
I want to prove that if all the eigenvalues of a linear transformation T : V --> V are zero, then T = 0. I think this is obvious but I'm having difficulty putting it into words.
If all the eigenvalues of T are zero, then there exists a basis B for V in which [itex][T]_B [/itex] is the zero matrix. Thus [itex][T(v)]_B=[T]^B_B[v]_B = 0\; \forall v \in V[/itex].
I think I should show that the matrix representation of T in an arbitrary basis D is zero. We have from second year linear algebra [itex][id]^B_D [T]^B_B [id]^D_B = [T]^D_D[/itex].
Thus
[itex][T(v)]_D = [T]^D_D [v]_D = [id]^B_D [T]^B_B [id]^D_B [v]_B = 0[v]_D = 0[/itex].
But [itex]v \in V[/itex] is abritrary so [itex][T]^D_D [v]_D = 0 \; \forall [v]_D \in \mathbb{F}^n[/itex] where [itex]\mathbb{F}[/itex] is the base field. Hence T = 0.
Is this correct?
If all the eigenvalues of T are zero, then there exists a basis B for V in which [itex][T]_B [/itex] is the zero matrix. Thus [itex][T(v)]_B=[T]^B_B[v]_B = 0\; \forall v \in V[/itex].
I think I should show that the matrix representation of T in an arbitrary basis D is zero. We have from second year linear algebra [itex][id]^B_D [T]^B_B [id]^D_B = [T]^D_D[/itex].
Thus
[itex][T(v)]_D = [T]^D_D [v]_D = [id]^B_D [T]^B_B [id]^D_B [v]_B = 0[v]_D = 0[/itex].
But [itex]v \in V[/itex] is abritrary so [itex][T]^D_D [v]_D = 0 \; \forall [v]_D \in \mathbb{F}^n[/itex] where [itex]\mathbb{F}[/itex] is the base field. Hence T = 0.
Is this correct?
Last edited: