Problem 14. Prove that a function f: V→ W is linear if and only if its graph¹ is a subspace of VW.

3. Let V = P3, the vector space of polynomials of degree at most 3. Let B {1, t, t², t³} and C = {t³,t,t², 1} be bases for P3. If f(t) = (1+ t)(t² + 4t + 1). Find [f(t)]8 and [f(t)]c

10.Suppose that each of the vectors x(1), …, x(m) has n components, where n < m. Show that x(1), …, x(m) are linearly dependent. In each of Problems 11 and 12, determine whether the members of the given set of vectors are linearly independent for −∞ < t < ∞ . If they are linearly dependent, find the linear relation among them.

3. In M3x2(F), prove that the set { ( ) ( ) ( ) ( ) ( ) } " 0 0 1 is linearly dependent.

Let V and W be vector spaces and let T : V → W be a linear transforma- tion. Suppose that {T(v₁), T(v₂),..., T(vn)} is linearly independent. Prove that {V₁, V2,..., Vn} is linearly independent.