Linear Transformations and Bases

I need some help or at least some assurance that my thinking on linear transformations and their matrix representations is correct.

I assume when we specify a linear transformation eg F(x,y, z) = (3x +y, y+z, 2x-3z) for example, that this is specified by its action on the variables and is not with respect to any basis.

However when we specify the matrix of a linear transformation T: V --> W that this is with respect to a basis in V and a basis in W

Of course if we have a linear transformation S: V -->V it could be that the two bases are the same.

If we are dealing in Euclidean space and no basis is mentioned regarding the matrix of a linear transformation, then I am assuming the standard bases are assumed.

Can someone either confirm I am correct in my thinking or point out the errors in my thinking?

Peter

Re: Linear Transformations and Bases

Regarding your example F(x,y, z) = (3x +y, y+z, 2x-3z), consider the following.

Let , and . Then .

Obviously or .

Standard canonical basis in , vectors , , are mapped to , columns of the matrix :

, and .

Every two bases in vector space V are equivalent. Meaning that every vector can be given in either one of them as a linear combination of the vectors forming the basis, the difference is in coefficients used.

If nothing is specified then standard basis is assumed.

Re: Linear Transformations and Bases

in general, the matrix of a linear transformation is determined "only up to similarity" (if T is a linear transformation, and A is the matrix for T in the standard basis, then if P is the (necessarily invertible) "change-of basis" matrix from the standard basis to any other basis, the matrix for T in the NEW basis is: PTP^{-1}).

this is why similarity classes are important in the study of matrices.

note that just WRITING a vector as (x,y,z) is actually implying a basis:

(x,y,z) = x(1,0,0) + y(0,1,0) + z(0,0,1) = xe_{1} + ye_{2} + ze_{3}.

that is, in the standard basis B = {e_{1},e_{2},e_{3}} the "coordinates" of a vector (x,y,z) are just [x,y,z]_{B}.

in R^{n}, particularly for n = 2 and n = 3, the normal "coordinate axes" are so firmly entrenched in our minds, we forget we PUT them there in the first place, there is no "x-axis" in actual space. so, for a rotation (like, for example, a ferris wheel), we "choose" an origin (the center of the wheel) to be able to describe its motion mathematically. it's sort of like choosing a language to give expression to an idea....the idea is just what it is, but we need WORDS to convey it.

Re: Linear Transformations and Bases

Thanks for the help above

Much appreciated

Peter