Column Space - Definition

Definition

Let A be an m × n matrix, with column vectors v1, v2, ..., vn. A linear combination of these vectors is any vector of the form

where c1, c2, ..., cn are scalars. The set of all possible linear combinations of v1,...,vn is called the column space of A. That is, the column space of A is the span of the vectors v1,...,vn.

Example
If, then the column vectors are v1 = (1, 0, 2)T and v2 = (0, 1, 0)T.
A linear combination of v1 and v2 is any vector of the form
The set of all such vectors is the column space of A. In this case, the column space is precisely the set of vectors (x, y, z) ∈ R3 satisfying the equation z = 2x (using Cartesian coordinates, this set is a plane through the origin in three-dimensional space).

Any linear combination of the column vectors of a matrix A can be written as the product of A with a column vector:

Therefore, the column space of A consists of all possible products Ax, for x ∈ Rn. This is the same as the image (or range) of the corresponding matrix transformation.

Read more about this topic:  Column Space

Famous quotes containing the word definition:

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)

    “The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
    —William James (1842–1910)

    “It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
    —François, Duc De La Rochefoucauld (1613–1680)