Description
Programming Preamble:
Matlab: R=rand(5,7) produces a 5×7 matrix with random entries.
Matlab: A’ produces A transpose. Changes column vectors to row vectors.
Matlab: rref(R)produces the reduced row echelon form of the matrix.
Matlab: cat(2,A,B) concatenation of A and B (use to produce an augmented matrix).
python: import numpy
R=numpy.random.rand(5,7)
Computing Assignment
Required submission: 1 page PDF document with your answers to the problems here, and 1 page
PDF document with your Matlab or Python code, both uploaded to Crowdmark (so, upload 2
pages).
1. Part 1 – Solutions of systems of linear equations
• By considering random matrices of appropriate sizes, find “emperical evidence” that
substantiate the following statements. (In this part, m and n are both integers and both
are greater than 6, so 7 or larger).
(a) A system of n linear equations in n unknowns typically has a unique solution.
(b) A system of m linear equations in n unknowns, where m > n, typically has no
solution.
(c) A system of m linear equations in n unknowns, where m < n, typically has many
solutions.
• Give examples of exceptions for each case (but for this part, you can use integers m and
n that need only be larger than 2, so 3 or larger).
1
2. Part 2 – Linear independence, Intersection of subspaces
• Consider the set of vectors in R
5
;
B1 = {w1, w2, w3, w4, w5, w6}
where
w1 =
−1
1
2
4
1
, w2 =
−1
1
2
1
1
, w3 =
3
1
−1
2
0
, w4 =
2
1
0
−1
, w5 =
5
4
1
11
3
, w6 =
1
0
−1
2
1
Show B1 is a linearly dependent set. Then, demonstrate the conclusion of Theorem
1.2.2: Find a maximal linearly independent set B′
1
of vectors from B1, and show that
the vectors from B1 that are NOT in B′
1
set are contained in the span of B′
1
(and hence,
that span B1 = span B′
1
).
What is the dimension of span B1?
• Consider the set
B2 = {z1, z2, z3, z4, z5}
where
z1 =
5
2
1
7
1
, z2 =
2
−1
0
0
1
, z3 =
1
2
1
1
0
, z4 =
2
−4
−2
4
1
, z5 =
0
1
2
3
−1
• Find all the vectors in the intersection span B1 ∩ span B2. Show that this is a subspace.
2
