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:
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)