Can every vector be written in linear combination of eigenvectors with distinct eigenvalues, when $\mathrm{T}$ is diagonalizable?

Question

If $\mathrm{T}$ is a linear operator on $\mathrm{V}$, and $\mathrm{T}$ is also diagonalizable, does it hold that,

Every vector of $\mathrm{V}$ can be written in a linear combination of eigenvectors of $\mathrm{T}$ with distinct eigenvalues.

I know that every vector can be written in a linear combination of eigenvectors, if diagonalizable. However, I'm not sure whether it still holds when the eigenvectors are with distinct eigenvalues.

Answer 1

(If $T$ is diagonalizable), every vector is even a sum of eigenvectors with distinct eigenvalues: if $v=∑_ix_iv_i$ with $v_i$ eigenvector for eigenvalue $λ_i$, then $v=∑_λu_λ$ with $u_λ=∑_{λ_i=λ}x_iv_i$, i.e. you group within $u_λ$ all $x_iv_i$'s corresponding to the eigenvalue $λ$.