Homework 6, due November 15, 2006.

[goodwine]

If you are asked to justify your answer you must prove the assertion. Note that to prove something is not true, a counter-example will suffice.

1. Show that the following sets produce real linear spaces when appropriate operations, i.e., scalar multiplication and vector addition, are defined on them.
   1. The set of real m by n matrices of the form
      • 
      where each element is a real number.
   2. The set of real polynomials of degree less than or equal to n in a real variable t. Thus, a typical element of this set is of the form
      • 
      where the coefficients and t are real numbers.
2. Let M(33) be the set of real 3 by 3 matrices. Let Sym(M(33)) be the subset of matrices which are symmetric. Thus
   • 
   is in Sym(M(33)) if a_ij=a_ji for i,j=1,2,3. Also, let Skew(M(33)) be the subset of matrices which are skew-symmetric. Thus
   • 
   is in Skew(M(33)) if b_ij=-b_ji for i,j=1,2,3. Show that Sym(M(33)) and Skew(M(33)) are subspaces of M(33).
3. Prove the following elements of a linear vector space are unique:
   1. the zero element; and
   2. the additive inverse (recall that for x in a linear space the additive inverse is -x such that x+(-x)=0).
4. Is the set M(mn) (from problem 1) where vector addition is defined as the usual matrix multiplication a vector space? Justify your answer.
5. Is the set M(mn) with vector addition defined by x+y=0 for x,y elements of M(mn) and 0 being the m by n matrix with all zero elements a linear vector space? Justify your answer.
6. Let U and V be subspaces of a vector space X.
   1. Is the intersection of U and V a subspace of X? Justify your answer.
   2. Is the union of U and V a subspace of X? Justify your answer.

_________________
I would rather discover a single fact, even a small one, than debate the great issues at length without discovering anything at all.
--- Galileo Galilei.