Page 368 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

22. (For Readers Who Have Studied Calculus) Show that the set of continuous functions on such that

is a subspace of .

Indicate whether each statement is always true or sometimes false. Justify your answer by
23. giving a logical argument or a counterexample.

(a) If is any consistent linar system of m equations in n unknowns, then the

solution set is a subspace of .

(b) If W is a set of one or more vectors from a vector space V, and if is a vector in

W for all vectors u and v in W and for all scalars k, then W is a subspace of V.

(c) If S is a finite set of vectors in a vector space V, then span(S) must be closed under
addition and scalar multiplication.

(d) The intersection of two subspaces of a vector space V is also a subspace of V.

(e) If , then .

24.
(a) Under what conditions will two vectors in span a plane? A line?

(b) Under what conditions will it be true that ? Explain.

(c) If is a consistent system of m equations in n unknowns, under what conditions

will it be true that the solution set is a subspace of ? Explain.

Recall that lines through the origin are subspaces of . If is the line is the line

25. , is the union a subspace of ? Explain your reasoning.

26. be the vector space of matrices. Find four matrices that span .
(a) Let

(b) In words, describe a set of matrices that spans .