Math 232 – Computing Assignment 1

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

3

−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

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