Question

what is meant by basis vectors?

Answer 1

basis vectors for a vector space (or subspace) of dimension 'n' is a set of n vectors whose span is the entire space. In other words, every vector in the space is a linear combination of the basis vectors.
For example in the Cartesian plane (R^2), one basis could be the vectors (1,0), and (0,1). Any vector (x,y) = x(1,0) + y(0,1).

Answer 2

there is more to it than that, but this should at least provide an understanding.

Answer 3

Cartesian plane (R^2); by this you mean its like a disc right?

Answer 4

The xy-plane, what we usually graph on all the time.

Answer 5

Spanning is only one of the conditions.

Answer 6

A basis has two conditions.
1) It must span the vector space
2) It must be linearly independent

Answer 7

right right. i dont wish to confuse the masses though lol.

Answer 8

I like my bases to be orthogonal, lol.

Answer 9

Yes orthogonal bases are very nice.

Answer 10

With an orthogonal basis, if you want to express an arbitrary vector as a linear combination of the basis, all you need to do is compute some dot products.

Answer 11

On the other hand if the basis is not orthogonal you need to solve a system of equations.

Answer 12

@alchemista: what do you mean when you say that its linearly independent?

Answer 13

I like ab = a dot b + a wedge b (which can be generalized to multivectors).

Answer 14

guys i am a total newbie at this...i have no clue what terminologies you guys are using.. whats orthogonal basis?

Answer 15

xy plane, 1 unit on x, 1 unit on y is a basis.

Or xyz in 3d.

Perpendicular to each other.

Answer 16

@estudier: i understand what a dot b is... but whats a wedge b??

Answer 17

thanx estudier i get the idea what ortho basis is

Answer 18

In 3D it calculates a value equivalent to the cross product (which represents the area of the implicit //gram formed by the vectors a and b).

Difference is cross only works in 3D, wedge works all dim and orientation is automatic.

Answer 19

A set of vectors are linearly independent if no vector in the set can be expressed as a linear combination of other vectors in the set.

In other words you have the condition that for a set of vectors \[B=\{v_1, v_2, ..., v_n\}\]
For coefficients \[\{a_1, a_2, ..., a_n\}\]
If the set is linearly independent
\[a_1v_1 + a_2v_2 + ... + a_nv_n = 0\]
Only when a_i = 0 for all i=0 to n.

Answer 20

You can therefore think of a wedge b as a "patch" of oriented area (of any shape).

Answer 21

okay got it...can you provide a mathematical example of how wedge works?
n thanx alchemista i undrstood what you said.

Answer 22

@estudier: what do tou mean by this:You can therefore think of a wedge b as a "patch" of oriented area (of any shape)?

Answer 23

*you

Answer 24

a wedge b = (a1b2 -a2b1)e12 + (a2b3-a3b2)e23 + (a3b1-a1b3)e31

(e_i is a (orthogonal) unit vector)

The bracketed represent areas of //grams projected on to the basis planes.

Compare with the formula for the cross.

Answer 25

hmm...except for the e_ij and a minus sign in the middle expression it looks just like a cross product. do i have it right?

Answer 26

Yes, these apparently minor differences are what allows the generalization.

In fact there is an entire separate algebra for the wedge, the exterior algebra

http://en.wikipedia.org/wiki/Exterior_algebra

However, for many practical applications, this "outer product" works best when combined with the inner ("dot") product 8because ab is invertible).

Answer 27

okay thanx a lot