Howto define the Column space of nxn matrix

[Susanne217]

Homework Statement

I thought that if you have a square matrix then the column space is the set of all linear independent vectors which can be written as a linear combinations of the others? Which inturn is the same as range of the Matrix?
Am I wrong?

 

[VeeEight]

The column space of a matrix is the set of all possible linear combinations of its column vectors.
It is the same as the range for the corresponding transformation matrix.

 

[Susanne217]

The column space of a matrix is the set of all possible linear combinations of its column vectors.

and that's the same as the range of the matrix?

 

[VeeEight]

Yes, it is the same as the range of the matrix (for the corresponding transformation).

 

[Susanne217]

Yes, it is the same as the range of the matrix (for the corresponding transformation).

It little follow-up question the same n x n matrix B is a 6 x 6 then then following how do I show if the matrix equation Bx = c has a solution for all [tex]c \in \mathbb{R}^6[/tex] is that if all the columns in B are linear independent and thusly spans the whole of [tex] \mathbb{R}^6[/tex] and then claim is true? and if they don't the claim is false for the matrix B?