Maths Not Boring: Reasons & Solutions

[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.