Coordinate Vector - Infinite Dimensional Vector Spaces

Infinite Dimensional Vector Spaces

Suppose V is an infinite dimensional vector space over a field F. If the dimension is κ, then there is some basis of κ elements for V. After an order is chosen, the basis can be considered an ordered basis. The elements of V are finite linear combinations of elements in the basis, which give rise to unique coordinate representations exactly as described before. The only change is that the indexing set for the coordinates is not finite. Since a given vector v is a finite linear combination of basis elements, the only nonzero entries of the coordinate vector for v will be the nonzero coefficients of the linear combination representing v. Thus the coordinate vector for v is zero except in finitely many entries.

The linear transformations between (possibly) infinite dimensional vector spaces can be modeled, analogously to the finite dimensional case, with infinite matrices. The special case of the transformations from V into V is described in the full linear ring article.

Read more about this topic:  Coordinate Vector

Famous quotes containing the words infinite, dimensional and/or spaces:

    In the order of literature, as in others, there is no act that is not the coronation of an infinite series of causes and the source of an infinite series of effects.
    Jorge Luis Borges (1899–1986)

    I don’t see black people as victims even though we are exploited. Victims are flat, one- dimensional characters, someone rolled over by a steamroller so you have a cardboard person. We are far more resilient and more rounded than that. I will go on showing there’s more to us than our being victimized. Victims are dead.
    Kristin Hunter (b. 1931)

    Le silence éternel de ces espaces infinis m’effraie. The eternal silence of these infinite spaces frightens me.
    Blaise Pascal (1623–1662)