Shakes Me Up: Range of a matrix (Column Space)

In linear algebra, the column space of a matrix (sometimes called the range of a matrix) is the set of all possible linear combinations of its column vectors. The column space of an m × n matrix is a subspace of m-dimensional Euclidean space. The dimension of the column space is called the rank of the matrix.

The column space of a matrix is the image or range of the corresponding matrix transformation.
The dimension of the column space is called the rank of the matrix.

Definition:

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

$c_1 \textbf{v}_1 + c_2 \textbf{v}_2 + \cdots + c_n \textbf{v}_n\text{,}$

where c₁, c₂, ..., c_n are scalars. The set of all possible linear combinations of v₁,...,v_n is called the column space of A. That is, the column space of A is the span of the vectorsv₁,...,v_n.

Basis

The columns of A span the column space, but they may not form a basis if the column vectors are not linearly independent. Fortunately, elementary row operations do not affect the dependence relations between the column vectors. This makes it possible to use row reduction to find a basis for the column space.

The left null space of A is the set of all vectors x such that x^TA = 0^T. It is the same as the null space of the transpose of A. The left null space is the orthogonal complement to the column space of A.

This can be seen by writing the product of the matrix $A T$ and the vector x in terms of the dot product of vectors:

$A^T\textbf{x} = \begin{bmatrix} \textbf{c}_1 \cdot \textbf{x} \\ \textbf{c}_2 \cdot \textbf{x} \\ \vdots \\ \textbf{c}_n \cdot \textbf{x} \end{bmatrix}\text{,}$

Shakes Me Up

Saturday, December 3, 2011

Range of a matrix (Column Space)

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