18.6. EXERCISES 403

18.6 Exercises1. Let {u1, · · · ,un} be vectors in Rn. The parallelepiped determined by these vectors

P(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 {Ax : x ∈ P(u1, · · · ,un)} is also a paral-lelepiped.

2. In the context of Problem 1, draw P(e1,e2) where e1,e2 are the standard basis vec-

tors for R2. Thus e1 = (1,0) ,e2 = (0,1) . Now suppose E =

(1 10 1

)where E

is the elementary matrix which takes the second row and adds to the first. Draw{Ex : x ∈ P(e1,e2)} . In other words, draw the result of doing E to the vectors inP(e1,e2). Next draw the results of doing the other elementary matrices to P(e1,e2).

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

(a)(

1 2 00 1 7

)

(b)

 1 0 0 00 0 1 20 0 0 0

(c)

 1 1 0 0 0 50 0 1 2 0 40 0 0 0 1 3



4. Row reduce the following matrices to obtain the row reduced echelon form. List thepivot columns in the original matrix.

(a)

 1 2 0 32 1 2 21 1 0 3



(b)

1 2 32 1 −23 0 03 2 1



(c)

 1 2 1 3−3 2 1 03 2 1 1



5. Find the rank of the following matrices. If the rank is r, identify r columns in theoriginal matrix which have the property that every other column may be written asa linear combination of these. Also find a basis for column space of the matrices.

(a)

1 2 03 2 12 1 00 2 1

 (b)

1 0 04 1 12 1 00 2 0

