Understanding Change of Basis in Vector Spaces

[princy]

hi.. can anyone say what is the concept behind change of basis.. y do we change a vector of one basis to another?

 

[Stephen Tashi]

In general, each component of a vector in one basis becomes a linear combination of the components of the vectors in the other basis.

 

[wisvuze]

One reason why we change bases because we are looking for another representation of some vector. Representing a vector as a vector generated by a basis gives us a concrete interpretation of this vector, and if we were ever given an "easy" basis, we can give ourselves an "easy" representation of a vector.
For example, suppose you had a basis {v_1, ... v_n } so that T ( v_i ) = c_i v_i where c_i is some constant and v_i is anything in your basis. Then any vector in your space, w, can be written as w = a1v1 + a2v2 +... . + an vn, and Tw = a1 Tv1 + ... an Tvn = a1 c1 v1 +... an cn vn
So all T does to some vector w in V is scale the coefficients by another factor
the matrix interpretation would be that if you had a n x n diagonal matrix, and an n x 1 column vector, to multiply those two, you'd only have to multiply each row by whatever is on the corresponding row on the matrix ( whatever is on the diagonal )

 

[Deveno]

yes, some bases make calculations easier, a matrix might have a "nicer" form, such as upper triangular, or diagonal, or one might want to make a basis orthonormal, to simplify calcuating inner products.

another reason might be that you have, for example, some data that is given in terms of certain linearly independent functions, but you want to express these in terms of "standard functions". perhaps "cost" is determined by one polynomial, and "productivity" by another polynomial, and you want to express the results in terms of 1,x,x^2,x^3, etc.

sometimes, one basis makes the geometry more transparent, and the spatial relationships more obvious. you might transform a "slanted" space, to one that has perpendicular axes, to get a better feel for what things "look like".

 

[td21]

Beware that if you want to express a linear transformation by a matrix in some "weird" space, you have to include the change of basis matrix, which is the identity transformation of vectors between weird space and Euclidean space.

For example,
S=(s1,s2,s3) is the one that identity transform matrix from S-space to Euclidean 3D space.

But what is the matrix from Euclidean 3D space to S-space? It is S inverse!