Quiz 5 Ma 322 005 Name ______________

1. Find a basis for the column space of A = [Maple Math] What is the dimension of N(A)?

By inspection, the first and third columns of A are the pivot columns and so form a basis for the column space. So the dimension of C(A) is 2. Since dim(C(A))+dim(N(A)) = #cols of A = 3, we get that dim(N(A))=1. (Note: this could also be computed directly by finding a basis for N(A).

2. Suppose that V is a vector space with basis [Maple Math] . Let [Maple Math] and w = [Maple Math] . Show that u and w are linearly independent.

Write s u + t w = O and show that s = 0 = t.

Well s u + t w = s ( [Maple Math] ) + t ( [Maple Math] ) = [Maple Math] = O. Since [Maple Math] and [Maple Math] are linearly independent, [Maple Math] and [Maple Math] . Adding, we get [Maple Math] , so s = 0. Substitute s = 0 into 1st eqn and get t = 0. Hence u and w are linearly independent.

3. Let [Maple Math] and [Maple Math] . Explain why [Maple Math] and [Maple Math] cannot form a basis for [Maple Math]

The dimension of [Maple Math] is 4 and hence any basis for [Maple Math] must have 4 vectors.