14-240/Tutorial-October14

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Boris

Elementary and (Not So Elementary) Errors in Homework

(1) Bad Notation


Let be matrices.


We want to equate to the set of all symmetric matrices. Here is the wrong way to write this:


.


Firstly, is the set of all linear combinations of . To equate it to a single

symmetric matrix makes no sense. Secondly, the elements are undefined. What are they suppose to

represent? Rational numbers? Real numbers? Members of the field of two elements? The following way of writing erases those issues:


where is an arbitrary field.


(2) Algorithm vs. Proof

When solving a problem that requires a solution to a linear equation, it is not always obvious which of the following you

should show:

a) An algorithm for finding the solution
b) A proof that a solution is correct

If the problem asks to solve a linear equation, then just show (a). Otherwise, for problems such as this:

Determine if the vector is a linear combination of the vectors in .

Show both (a) and (b) to be on the safe side.

Problem 5h) in Homework 3 for all Fields

For an arbitrary field , determine if the matrix is in span .


Proof:

We show that .

We show that .
Assume that .
Let .
Then .
Then .
Since and the entries of the matrix are from , then .
Then .
Then .
We show that .
Assume to the contrary that .
Then .
Then this system of linear equations has a solution:
.
When solving this system, we see that it has no solution.
This contradicts the assumption that it has a solution.
Then .
Then . Q.E.D.

A Field Problem

A Dimension Problem

Nikita