how to find column space of a matrix

Question

Find the column space of matrix $A$ if $$ A= \begin{pmatrix}10 & 2 & 3 \\ 2 & 4 & 6 \\ 2 & 4 & 6 \end{pmatrix}$$

This is what I've done so far but I'm not sure if it is correct.

$$ \left[ \begin{array}{ccc|c} 10&2&3&x\\ 2&4&6&y\\2&4&6&z \end{array} \right] $$

reduced down to $$ \left[ \begin{array}{ccc|c} 10&2&3&x\\ 0&-18&-27&x-5y\\0&0&0&z-y \end{array} \right] $$

Does this mean the column space is $z - y = 0$?

Answer 1

Hint:

note that the first two columns of the matrix are linearly independent vectors, but the third is $3/2$ the second.

Answer 2

Do row reduction on the transpose of the given matrix to get the row space of the transpose which will inturn give you the column space of $A$.