IIUC, this is searching for f(X) = b, where b is not vector-like, but rather a Matrix.
Either try to define the missing orthogonalize! method for your output type,
or convert to a regular array (the collect used below) ?
julia> using KrylovKit
julia> b = rand(3, 3)
julia> res, info = linsolve(b) do x
return transpose(x) |> collect
end;
julia> transpose(res) ≈ b
true