Math 110—Linear Algebra
Fall 2009, Haiman
Problem Set 2
Due Monday, Sept. 14 at the beginning of lecture.
1. Section 1.5, Exercise 3.
2. Let Sbe the subset
{sin2(x),sin(2x),cos(2x),1}
of the vector space F(R,R). Which subsets of Sare linearly dependent and which are
linearly independent?
3. Prove that if S={v1, . . . , vn}is a finite, linearly independent set of vectors in a vector
space V, then every vector w∈Span(S) has a unique expression as a linear combination
a1v1+· · · +anvn.
4. Find a basis of the subspace of symmetric matrices in M3×3(R). What is the dimension
of this subspace?
5. Prove that if Vis a vector space over F2with finite dimension n, then Vis a finite set.
How many elements does it have?