Midterm 2 Solutions APPM 3310: Matrix Methods Page 1
Midterm 2 Solutions
1. Consider A=
1 5 0 −3
2 10 1 −4
−1−5 1 5
. Use this definition of Afor all parts of this problem.
Give a basis for (a) the row space of A, (b) the column space of A, (c) the null space of A, (d)
the left null space of A.
Row reduction gives A∼U=
150−3
0 0 1 2
0 0 0 0
.A basis for the row space is the first two
nonzero rows of U(or A), that is,
1
5
0
−3
,
0
0
1
2
. For the column space, use the pivot columns
(the first and third columns) of A:
1
2
−1
,
0
1
1
. (Note: zero points for using the first and
third columns of U.) For the null space, the columns 2 and 4 have free variables, leading to a basis
set
−5
1
0
0
,
3
0
−2
1
. To find the left null space, row reduce AT∼
1 2 −1
0 1 1
0 0 0
0 0 0
.The third
column corresponds to a free variable, so the basis vector is
3
−1
1
.
(e) What is the rank of A? (f) What is the rank of AT?
Ahas two pivots, so its rank = 2. Similarly, AThas two pivots, so its rank is 2. The purpose of
this question was to help you find errors: you should remember that rank A= rank ATalways! So
if you made an arithmetic error and claimed the ranks were different, you did not receive full credit.
2. You are given 4 vectors, x1,x2,x3,x4∈R4. This set spans a d-dimensional subspace of R4.
(a) When, if ever, is d < 1?
When all 4 vectors are the zero vector; when the matrix A= [x1x2x3x4] is the zero matrix.
(b) When, if ever, is d= 1?
When all 4 vectors point along the same line; when all 4 vectors are multiples of each other; when
the set has only one linearly independent vector; when the matrix A= [x1x2x3x4] is rank 1 (one
pivot).
(c) When, if ever, is d= 2?
When all 4 vectors lie in a same plane; when the set has only two linearly independent vectors;
when the matrix A= [x1x2x3x4] is rank 2 (two pivots).
(d) When, if ever, is d= 3?
When the set has only three linearly independent vectors; when the matrix A= [x1x2x3x4] is
rank 3 (three pivots).
(e) When, if ever, is d= 4?
When the vectors are linearly independent; when the matrix A= [x1x2x3x4] is full rank (four
pivots).
(f) When, if ever, is d > 4?
Never: it is impossible for vectors ∈R4to span a space with dimension >4.
