**Math 3592H Honors Mathematics I Fall Semester 2005
**

Exercises:

Section 2.5 (pages 221-224): 1, 2, 3, 4, 6, 6b*, 7, 8*, 9*, 10*, 13, 15*, 19, 20

Extra questions:

1. Construct a matrix with no zero entries whose echelon form is

.

2. Extend each of the following lists of vectors to a basis of the whole space (I write row vectors instead of column vectors because they fit on lines better):

(i) (1,1,1). (ii) (1,2,3), (0,2,3). (iii)* (1,2,3,4), (1,1,1,1), (0,0,1,2).

3. Consider the vectors (1,2,3), (3,2,1), (1,0,–1), (–1,2,5).

(i) Find a subset of these vectors which is a basis for the space which they span, and which contains the first vector.

(ii)* Find a subset of these vectors which is a basis for the space which they span, and which contains the last vector.

4*. Let U be the subspace of 4-dimensional space which consists of the vectors (w,x,y,z) for which w + x + y + z = 0. Extend the vector (1, –1, 1, –1) to a basis of U.

5*. Let U and V be linear subspaces of 10-dimensional real space. Suppose that U is a subset of V and that dim U = dim V = 5. Prove that U = V.