# Homework Help: Howto define the Column space of nxn matrix

1. Feb 8, 2010

### Susanne217

1. The problem statement, all variables and given/known data

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?

2. Feb 8, 2010

### 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.

3. Feb 8, 2010

### Susanne217

and thats the same as the range of the matrix?

4. Feb 8, 2010

### VeeEight

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

5. Feb 8, 2010

### Susanne217

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 $$c \in \mathbb{R}^6$$ is that if all the columns in B are linear independent and thusly spans the whole of $$\mathbb{R}^6$$ and then claim is true? and if they don't the claim is false for the matrix B?