## 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.

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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.

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