# Basis for such a quotient vector space

Hi all,

Here is my code:

`A = matrix(QQ, 4, 3, [1, 2, 3, 5, 10, 15, 0, 0, 0, -2, -4, -6])`

`B = matrix(QQ, 5, 4, [0, 2, 7, 5, 0, 0, 0, 0, 0, -2, -7, -5, 0, 4, 14, 10, 0, 6, 21, 15])`

`W = A.transpose().image()`

`V = B.right_kernel()`

`V/W`

This way W is the range of the linear map $$A:\mathbb{Q}^3\longrightarrow\mathbb{Q}^4,$$ V is the nullspace of $$B:\mathbb{Q}^4\longrightarrow\mathbb{Q}^5,$$ Since BA=0, W is a subspace of V. and the command V/W returns in particular the dimension of the quotient vector space V/W which is equal to 2.

What I want is a basis of V/W, i.e. 2 vectors of $\mathbb{Q}^4$ which are in V, in the form of a 2x4 matrix the same way V.basis_matrix() returns a basis for V.

How can I do this?

Thanks!