124 CHAPTER 4. ROW OPERATIONS

4.6 Exercises

1. Let {u1, · · · ,un} be vectors in Rn. The parallelepiped determined by these vectorsP (u1, · · · ,un) is defined as

P (u1, · · · ,un) ≡

{n∑

k=1

tkuk : tk ∈ [0, 1] for all k

}.

Now let A be an n× n matrix. Show that

{Ax : x ∈ P (u1, · · · ,un)}

is also a parallelepiped.

2. In the context of Problem 1, draw P (e1, e2) where e1, e2 are the standard basis vectorsfor R2. Thus e1 = (1, 0) , e2 = (0, 1) . Now suppose

E =

(1 1

0 1

)where E is the elementary matrix which takes the third row and adds to the first.Draw

{Ex : x ∈ P (e1, e2)} .In other words, draw the result of doing E to the vectors in P (e1, e2). Next draw theresults of doing the other elementary matrices to P (e1, e2).

3. In the context of Problem 1, either draw or describe the result of doing elementarymatrices to P (e1, e2, e3). Describe geometrically the conclusion of Corollary 4.3.7.

4. Consider a permutation of {1, 2, · · · , n}. This is an ordered list of numbers taken fromthis list with no repeats, {i1, i2, · · · , in}. Define the permutation matrix

P (i1, i2, · · · , in)

as the matrix which is obtained from the identity matrix by placing the jth column of Ias the ithj column of P (i1, i2, · · · , in) . Thus the 1 in the ithj column of this permutation

matrix occurs in the jth slot. What does this permutation matrix do to the columnvector (1, 2, · · · , n)T ?

5. ↑Consider the 3 × 3 permutation matrices. List all of them and then determine thedimension of their span. Recall that you can consider an m× n matrix as somethingin Fnm.

6. Determine which matrices are in row reduced echelon form.

(a)

(1 2 0

0 1 7

)

(b)

 1 0 0 0

0 0 1 2

0 0 0 0

(c)

 1 1 0 0 0 5

0 0 1 2 0 4

0 0 0 0 1 3

