How to show one set of vectors span R3 if its components do?

Question

The question states

Show that if a, b, c are vectors in R3, then {b+c, c+a,a+b} spans R3 iff {a,b,c} spans R3.

I've started trying Gauss-Jordan with the first set in Ax = 0 form, but I haven't got anywhere.

What am I supposed to do from here?

Answer 1

HINT

Let consider

$$k_1(b+c)+k_2(c+a)+k_3(a+b)=0$$

$$(k_2+k_3)a+(k_1+k_3)b+(k_1+k_2)c=0$$

Answer 2

Hint

$\text{{a,b,c}}$ spans $\mathbb R^3$ implies that $l_1a+l_2b+l_3c=0 $ for some $l_1,l_2,l_3$. Now evaluate $l_1(b+c)+l_2(c+a)+l_3(a+b) $ and use the equality above.

Answer 3

Find a matrix $M$ such that $$\begin{bmatrix}\mathbf b+\mathbf c & \mathbf c+\mathbf a & \mathbf a+\mathbf b\end{bmatrix} = \begin{bmatrix}\mathbf a & \mathbf b & \mathbf c\end{bmatrix} M$$ and use properties of determinants to argue that the columns of the matrix on the left-hand side are linearly independent iff $\mathbf a$, $\mathbf b$ and $\mathbf c$ are.