# Thread: Column Space and Row space

1. ## Column Space and Row space

I have a n by n matrix in which i have to calculate the orthogonal basis for the column space and row space.
I keep getting the same basis for both. Is that supposed to happen when dealing with a square matrix, or am i doing something wrong?

here is the matrix :

1 4 0
2 8 1
1 4 0

2. Originally Posted by Carl505
1 4 0
2 8 1
1 4 0
We use theorem that the row space of two row equivalent matrices is the same.

Thus, we bring this matrix to row reduced form.

$\displaystyle \left[ \begin{array}{ccc}1&4&0 \\ 0&0&1 \\ 0&0&0 \end{array} \right]$

We easily see that $\displaystyle [1 \ 4 \ 0] \mbox{ and }[0 \ 0 \ 1]$ are linearly independent after we drop the last row. Thus, those two form a basis for the row space.

Now the question is what about the coloum space? The theorem says that it is the correspoding to the coloum vectors.

If we look at the row reduced matrix we see that $\displaystyle \left[ \begin{array}{c}1\\0\\0 \end{array} \right] \mbox{ and }\left[ \begin{array}{c}0\\1\\0 \end{array} \right]$ are linearly independent because the second one can be obtained from multiplication of the first one by 4.

Now the corresponding vectors in the original matrix are:
$\displaystyle \left\{ \left[ \begin{array}{c} 1\\2\\1 \end{array} \right] , \left[ \begin{array}{c}0\\1\\0 \end{array} \right] \right\}$