Linear Transformations Mock Tests

Let V be the vector space of all polynomials over ℝ of degree at most 3. Let T: V → V be the linear transformation defined by T(p(x)) = p′(x) + p(x). The eigenvalues of T are

Let T: R³ → R² be the linear transformation defined by T(x, y, z) = (x + y, y + z). The rank and nullity of T are respectively

Let T: ℝ³ → ℝ² be defined by T(x, y, z) = (x + y, y + z). The nullity of T is

Let T: V → V be a linear operator on a finite-dimensional vector space V. If every subspace of V is invariant under T, then T is

Let T: ℝ² → ℝ² be the linear transformation that rotates vectors counterclockwise by an angle θ about the origin. The determinant of T is