Maths Not Boring: Reasons & Solutions

[mathwonk]

but any two baSES are always equivalent, so you are just saying orthogonal bases exist.

 

[J77]

I think the poll is quite closed minded. Someone may say maths is dry and boring, and its not always because they don't understand it. In fact, I think that's quite a supremest statement to make! This discussion is being made in the General MATH section, so it is not exactly the most fair treatment of the matter, but as some of you know here, before my pure math days I studied Physics with a passion, I only got into pure Mathematics about a year ago. At that time, I just wasn't that interested in it. I studied it only because I needed it to advance my studies in Physics, and it was through this I found my interest. The point is, I Understood the maths just as good as anyone did, but I still wasn't very interested in it. To me, it was a tool for my physics studies, I'm sure an electrician doesn't find his screwdriver terribly interesting! Just because you don't find it interesting doesn't mean you don't understand it. I'm sure many of you hated doing essay's on poets in high school, I know i don't, however I still understand the syllabus and perform quite well on my tests. I find the subject dry and boring, but I understand the content fine.

What an excellent comment!

The best in this thread

 

[camilus]

yep it all depends on taste. Personally, I find most applied math dry and boring while I find pure math as mankind's greatest achievement as far as creativity, imagination, and intuition go.

But I guess this is a bias view considering that I am a platonist..

 

[chaoseverlasting]

Math requires more patience than most people would like to develop to do it properly. You can't just rush into it all the time. Sometimes, you need to get your head on right to do a problem. People just have a problem with that. Plus you have to want to do the problem in the first place.

 

[ice109]

but any two baSES are always equivalent, so you are just saying orthogonal bases exist.

if two bases span the same space they're equivalent?

 

[Hurkyl]

if two bases span the same space they're equivalent?

Sure. This is certainly a reasonable usage of the word 'equivalent'.

It's common in mathematics to look for generating sets for a structure. In this context, any set of vectors is a generating set for their span. Usually, the structure is the more interesting object of study, so it is common to define an equivalence relation that says two sets are equivalent if and only if they generate the same structure. In this case, two sets of vectors are equivalent if and only if they have the same span.

 

[ice109]

Sure. This is certainly a reasonable usage of the word 'equivalent'.

It's common in mathematics to look for generating sets for a structure. In this context, any set of vectors is a generating set for their span. Usually, the structure is the more interesting object of study, so it is common to define an equivalence relation that says two sets are equivalent if and only if they generate the same structure. In this case, two sets of vectors are equivalent if and only if they have the same span.

so then what is orthogonal truly? for some reason i think orthogonality is only relative to the coordinate system.

 

[slider142]

so then what is orthogonal truly? for some reason i think orthogonality is only relative to the coordinate system.

Orthogonality is only defined in inner product spaces, vector spaces which have an inner product defined on them. As such, it has little to do with linear independence, as that property does not require an inner product space. Orthogonality is relative to the inner product.

 

[Hurkyl]

so then what is orthogonal truly? for some reason i think orthogonality is only relative to the coordinate system.

When you have an inner product, then two vectors are orthogonal iff their inner product is zero.

For example, if I choose an origin on the Euclidean plane, then it naturally has a vector space structure. If I choose a unit length, then I can measure lengths. Then, I can define an inner product by

P \cdot Q = m\overline{OP} \; m\overline{OQ} \; \cos m\angle POQ.

Equivalently, if I set R = P + Q, then

P \cdot Q = \frac{m\overline{OR}^2 - m\overline{OP}^2 - m\overline{OQ}^2}{2}

(m\overline{AB} means the length of the line segment \overline{AB}. m\angle POQ means the angle measure of angle \angle POQ)

If you choose an orthogonal basis for the Euclidean plane, then the coordinate representation of the inner product is the dot product. But that's not true for skew bases.

 

[mathlove3.14159]

Me thinks math is good cause me can't do good english.