Math 327
Exam 5 - Practice Problems
1. Find a basis for the row space of each of the following matrices. Your basis should consist of rows of the original matrix.
(a)
1−2 5 4
2 1 −3 7
1−7 18 5
(b)
1 3 −4 1
0 4 −5 2
2−115
2 3 −4 7
3 2 −3 6
2. Find a basis for the column space of each of the matrices from problem 1. Your basis should consist of columns of the
original matrix.
3. Find the rank of each of the matrices from problem 1.
4. Find a basis for the null space of each of the matrices from problem 1.
5. Use the determinant to determine whether or not each of the following matrices has rank 3:
(a)
−231
0−1 2
2 1 6
(b)
−2 3 1
0 4 7
2 1 6
6. Use the rank of the coefficient matrix to determine whether or not each of the following homogeneous systems have a
non-trivial solution.
(a) A~x =~
0 if A=
5−124
2−301
1−537
0 2 −3 1
(b) A~x =~
0 if A=
5−1 2 4
2−3 0 1
1−5 3 7
0−7 6 13
7. Suppose that Ais a 4 ×7 matrix.
(a) What is the maximum rank of A?
(b) Could the columns of Abe linearly independent? Justify your answer.
(c) Could the rows of Abe linearly independent? Justify your answer.
(d) If the rank of Ais 3, find the nullity of A.
(e) If the rank of Ais 3, find the nullity of AT.
8. Let Abe an n×nmatrix. Show that rank A=nif and only if the columns of Aare linearly independent.
9. Consider the following functions from R3→R2:
L1
u1
u2
u3
=u1−u2
u3L2
u1
u2
u3
=u1−u2
0
L3
u1
u2
u3
=u1−u2
1L4
u1
u2
u3
=u1−u2
u2
3
(a) Determine whether or not each of these functions is a linear transformation. Justify your answer.
(b) For each Lithis is a linear transformation, find a matrix representing the linear transformation.
(c) For each Lithis is a linear transformation, find the kernel and range of the linear transformation.
10. Suppose that Lis a linear transformation with:
L
1
1
1
=
1
−1
0
,L
1
0
1
=
0
1
−2
, and L
0
0
1
=
3
0
0
