While doing real data analysis I came up with a problem. I have given lots of efforts to solve it and could not succeed.
Here is the problem:
Suppose, we have a set of vectors $X_1,X_2,\ldots,X_n$with dimension $1,\ldots,n$, respectively.
I want to define a notion of addition between two vectors of different dimensions which will be consistent with the usual vector addition in a n-dimensional vector space (that is componentwise addition of two same dimensional vector).
Here is some of my try:
Suppose $X=x_1$ and $Y=(y_1,y_2)'$ (prime denote transpose) than I define $X+Y=(x_1+y_1,x_1+y_2)'$.
Again if $X_1=x_1$ and $Y=(y_1,y_2,y_3)'$ then $X+Y=(x_1+y_1,x_1+y_2,x_1+y_3)'$
and if $X_1=(x_1,x_2)$ and $Y=(y_1,y_2)'$ then $X+Y=(x_1+y_1,x_2+y_2)'$ (usual vector addition).
But if $X_1=(x_1,x_2)$ and $Y=(y_1,y_2,y_3)'$ then how to define the addition between this two vectors such that this operation is consistent with the usual vector addition? And how to proceed for the other vectors of different dimensions.
Any idea or useful references will be gratefully appreciated.
Thanks in advance.
Best Answer
The easiest way to do this is to extend $X = (x_1, x_2)$ to $X' = (X_1, x_2, 0)$ (and otherwise just always extend vectors with as many zeroes as you need). The resulting space is infinite dimensional, and it is in some sense the "smallest" infinite-dimensional vector space construction there is. Polynomials, for instance, work this way if you ignore multiplication.