Column space

The columns of a matrix.

Image via Wikipedia

Let A be an m × n matrix, with column vectors v1v2, …, vn. 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 c1c2, …, 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 A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 2 & 0 \end{bmatrix}, then the column vectors are v1 = (1, 0, 2)T and v2 = (0, 1, 0)T.
A linear combination of v1 and v2is any vector of the form
c_1 \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix} + c_2 \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \\ 2c_1 \end{bmatrix}\,
The set of all such vectors is the column space of A. In this case, the column space is precisely the set of vectors (xyz) ∈ 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:

A\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix} = x_1 \textbf{v}_1 + \cdots + x_n \textbf{v}_n

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.

About NICO MATEMATIKA

Welcome to my blog. My name is Nico. Admin of this blog. I am a student majoring in mathematics who dreams of becoming a professor of mathematics. I live in Kwadungan, Ngawi, East Java. Hopefully in all the posts I can make a good learning material to the intellectual life of the nation. After the read, leave a comment. I always accept criticism suggestion to build a better me again .. Thanks for visiting .. : mrgreen:

Posted on October 8, 2011, in education and tagged , , , , , , , . Bookmark the permalink. Leave a comment.

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