Summary: LINEAR ALGEBRA (MATH 317H)
Assignment 11 - Dimension, Finite Dimensional spaces
(1) True or false. Give a counterexample in the case of a false statement. You
do not need to prove true statements.
(a) A vector space can have more than a basis.
(b) If V is a vector space having dimension n 1 then V has infinitely
many subspaces of dimension n - 1.
(c) The dimension of Pn is n + 1
(d) If a vector space has a basis then the number of vectors in every basis
is the same.
(e) The solution set of any homogeneous system of m equations in n un-
knowns is a subspace of Rn
(f) Any system of n linear equations in n unknowns has at least one solu-
(g) Any homogeneous system of n linear equations in n unknowns has at
least one solution
(2) Show that a subspace of a finite dimensional space is itself finite dimen-
(3) Prove that a linear independent system of vectors v1, . . . , vn in a vector