) Let V be a vector space over any field F: (a) Prove that V is infinite dimensional iff there is a sequence of vectors (V₁, V2, V3,...) in V such that V₁, V2, ..., Um is Linearly independent for each positive integer m.

Which of the following polynomials from the vector space R[x] is linearly expressed by the polynomials from the subset {1+x,1-x^2,x^3)? a.) x + x^2 + x^3 b.) 1+x+x^2 c.) 1+x+x^3 d.) 1 -x^2+x^3 e.) 1+2x+x^2

. Label the following statements as true or false, and give a brief justification for each:(a) A vector space cannot have more than one basis.(b) The dimension of P^n (the polynomials of degree n or less) is n

Let V be an n-dimensional vector space over the field of complex numbers C. Showthat V over R has dimension 2n.