Page 439 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(c) the X-matrix of size

In each part, show that the set of polynomials is a subspace of and find a basis for it.
10.

(a) all polynomials in such that

(b) all polynomials in such that

11. (For Readers Who Have Studied Calculus) Show that the set of all polynomials in that have a horizontal tangent
at is a subspace of . Find a basis for this subspace.

12. symmetric matrices.
(a) Find a basis for the vector space of all

(b) Find a basis for the vector space of all skew-symmetric matrices.

In advanced linear algebra, one proves the following determinant criterion for rank: The rank of a matrix A is r if and
13. only if A has some submatrix with a nonzero determinant, and all square submatrices of larger size have

determinant zero. (A submatrix of A is any matrix obtained by deleting rows or columns of A. The matrix A itself is also
considered to be a submatrix of A.) In each part, use this criterion to find the rank of the matrix.

(a)

(b)

(c)

(d)

Use the result in Exercise 13 to find the possible ranks for matrices of the form
14.