[Solved] MA502 Homework 1

1. Consider the space V of all vectors

{v = (v₁,,v_n) Rⁿ such that v = f(0) = (₁f(0),₂f(0),,_nf(0)) for some C¹ function f defined in a neighborhood of the origin}.

(1) Prove that V , equipped with the usual operations of vector sum and multiplication by a scalar is a vector space. (2) Prove that V = Rⁿ. 2. Show that if X and Y are subspaces of a vector space V, then X Y is

x₃

where the a_is and b_is are given real numbers.

• Prove that X and Y are vector spaces.
• Describe X Y in geometric terms, considering all possible choices of the coefficients. Is X Y a vector space?

4. Which of the following are subspaces of the given vector spaces? Justifyrigorously your answers. (1) {x Rⁿ : Ax = 0} Rⁿ, where A is a given m n
   • {p P : p(x) = p(x) for all x R} P, where P is the set of all polynomials with real coefficients.
   • {p P : p has degree less or equal than n} P.
   • {f C[0,1] : f(1) = 2f(0)} C[0,1], where C[0,1] is the set of all continuous functions on [0,1]. (5) The unit sphere in Rⁿ.

1

5. In the following, determine the dimension of each subspace and find abasis for it.

.

• The set of all n n square matrices with real coefficients that are equal to their transpose.

(3){p P2 : p(0) = 0} P2, where P2 is the set of all polynomials of degree 2.