Description
1) Matrix Rank. Let X =
1 0 1 0 −1
0 0 1 1 −1
0 0 0 0 0
0 0 0 0 0
.
a) What is the rank of X?
b) Find a set of linearly independent columns in X. Is there more than one set?
How many sets of linearly independent columns can you find?
c) A matrix A =
1 0 a
1 1 b
0 1 −1
. Find the relationship between b and a so that
rank{A} = 2. Hint: find a, b so that the third column is a weighted sum of the
first two columns. Note that there are many choices for a, b that result in rank 2.
2) Solution Existence.
A system of linear equations is given by Ax = b where A =
1 0
1 1
0 1
.
a) Suppose b =
8
6
−2
. Does a solution for x exist? If so, find x.
b) Suppose b =
4
6
1
. Does a solution for x exist? If so, find x.
c) Consider the general system of linear equations Ax = b. This equation says
that b is a weighted sum of the columns of A. Assume A is full rank. Use the
definition of linear independence to find the condition on rank A b that
guarantees a solution exists.
3) Non Unique Solutions.
a) Consider Ax = b where A =
1 −2
−1 2
−2 4
, b =
2
−2
−4
and x =
”
x1
x2
#
.
i) Does this system of equations have a solution? Justify your answer.
ii) Is the solution unique? Justify your answer.
iii) Draw the solution(s) in the x1-x2 plane using x1 as the horizontal axis.
b) If the system of linear equations Ax = b has more than one solution, then there
is at least one non zero vector w for which x + w is also a solution. That is,
A(x + w) = b. Use the definition of linear independence to find a condition on
rank{A} that determines whether there is more than one solution.
