1 of 1 If A is an n x n matrix, and x and y are different vectors in so that Ax=Ay, explain why this means that A cannot be invertible.Please remember to attach your solution as a .pdf file.Written Homework Instructions for MAT 343

Each week, there is a written homework assignment that is to be submitted online, through

blackboard. Do not post your answer as a text submission. Submit it as a PDF file and use

“attach file” to submit it.

Your solution document must be a PDF file, not a doc, docx or ods, text field or a jpg. It must be

typed, not handwritten.

Your solution document must contain your name and the homework/week number as a

header on every page. Do not put in your student ID, class number, class time, or other

identifying information.

Example: Jane Johnson – Written Homework Week #1

Do not copy the question text to your solution document. Just the answers including

supporting work.

If you violate the formatting guidelines in this document, you will lose points. In extreme cases,

the grader is permitted to refuse to grade your homework.

Important: you must show all work. If the question asks you to “explain” or similar, don’t treat

that part as optional. It is the most important part. Explanations must be written in full,

grammatically correct sentences and should be succinct.

Naked equations (i.e. equations without verbal explanation) do not explain anything.

Explanations of a general principle must also be general, i.e. apply to all instances of the

situation under consideration. Showing an example does not prove a general principle.

Composing your written homework

You have several software options for composing your written homework.

Microsoft Word

By far the easiest option is to use Microsoft Word. You will have to use the equation editor to

create equations. It is found on the INSERT tab on the right side. The editor has a convenient

interface that lets you create fractions, radicals, vectors and matrices and other mathematical

symbols easily.

Open Office/Libre Office Writer

Open Office/Libre Office can create mathematical equations too, but the editor is rudimentary

and unintuitive, and the formula language lacks good documentation. The functionality is

shamefully hidden under Insert/Object/Formula. I do not recommend this option.

LaTeX

A free alternative to MS Word that produces superior results for scientific documents is LaTeX.

It is the standard software for serious scientific typesetting, but requires a greater learning

investment. If you are thinking about pursuing graduate studies, you will be doing yourself a

favor by learning and mastering LaTeX. It’s either now or later.

There is a good free LaTeX tutorial at

http://mirrors.ctan.org/info/lshort/english/lshortletter.pdf

Under Windows, the standard LaTeX distribution is MiKTeX: http://miktex.org/

There is a 3rd party editor for Windows called WinEdt which seamlessly integrates with MiKTeX

and greatly simplifies and streamlines the process of creating LaTeX documents:

http://www.winedt.com/

All these systems can export files to PDF format, the format in which you are required to

submit your solutions.

No matter what system you use, you are expected to produce proper mathematical notation.

Avoid runaway equations and use appropriate line breaks. Do not typeset exponential

expressions as a^b, fractions like (a+1)/(b+1) or roots like sqrt(..). Do not represent matrices in

MATLAB form.

Examples of Good and Bad Solutions

Here is an example problem: if , are × matrices, = , ≠ , then must be

singular.

Example of a good solution:

If was regular, then we could left-multiply the equation = by −1 and obtain = .

That contradicts the assumption ≠ . Therefore, cannot be regular and must therefore be

singular.

Examples of bad solutions:

1. = → −1 = −1 → =

This solution is missing the logic of what’s going on, namely that we explore the

consequence of being regular and show that it leads to a contradiction with one of the

premises. Your reasoning needs to be explained in full sentences, not just implied.

2. If we let = (

1 0

1 0

) and = (

), then = and ≠ . is singular.

0 0

0 0

This solution shows an example. It illustrates the general principle, but it does not prove

the general validity of the principle.

1 0

1 0

) =(

), = , ≠ . singular. This solution combines the

0 0

0 0

mistakes of the first and the second example. It shows no explanations, just an example,

and it doesn’t even explain the example in complete sentences.

3. = (

4. If was regular, then we could left-multiply the equation = by ^-1 and obtain

= . That contradicts the assumption != I. Therefore, cannot be regular and must

therefore be singular.

While this solution is substantially correct, it does not use proper mathematical

typesetting.

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