Is adding k edges to spanning tree will result a graph with k cycle bases?

Question

The number of cycle bases of a connected graph with $n$ vertices and $m$ edges is $m - n + 1$. Let's say, $v_k$ is the number of cycle bases of a graph resulting from adding $k$ edges to spanning tree. Thus: $$v_k = (n-1+k)-n+1 =k $$ This result sort of confirming it. But this is not the proof. I was looking if there is a proof about this (preferably with a source I can look up)

Answer 1

Yes, this is true. One possible cycle basis is to pick a spanning tree, and for each non-tree edge, take the cycle consisting of that edge plus the path inside the tree from one of its endpoints to the other. (So if there are $k$ more edges in the graph than in the spanning tree, there are $k$ basis elements.)

This is called the fundamental cycle basis.