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# CS/ECE/ME532 Activity 4 solution

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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.
1 of 2
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.