Hello I have no idea where to start I really hope for your h

Hello) I have no idea where to start.... I really hope for your help here!!

let T: V --> V be a linear operator on a vector space of dimension 2. Assume that T is not a multiplication by a scalar. Prove that there is a vector v in V such that (v, T(v)) is a basis of V and describe the matrix of T with respect to that basis. Thanks in advance.

Solution

Since the dimension of V is 2, any linearly independent set of 2 elements forms a basis for V. In any linear space, two vectors are linearly dependent iff one of them is a scalar multiple of the other one, so if for all v V, {v, T(v)} is linearly dependent , then for all v V, T(v) = cv where c is a scalar. This is against the statement that T is not a multiplication by a scalar. Thus the set {v, T(v)} is linearly independent and therefore, forms a basis for V.