# Acyclic Chain Complex

I’m a little confused, for a chain to be acyclic, all Betti numbers must be zero. For a Betti number $$\beta$$

$$\beta_i=\dim(Z_i)-\dim(B_i)$$ where $$Z_i=\ker(\partial_i)$$ and $$B_i$$=im$$(\partial_i{+}_1))$$

I have $$\beta_1=\dim(Z_1)-\dim(B_1)=2-3$$, am I going about this the right way?

• Isn't the image of the arrow $\mathbb{R}^2\to\mathbb{R}^3$ $2$-dimensional, not $3$-dimensional, if I'm interpreting your indexing correctly? – Ben West Dec 4 '18 at 1:03
• Wouldn’t $B_1$=im$(T_2)=span{[1 -1 1]}$?, and that would be 3 dimensional? – LexyFidds Dec 4 '18 at 1:09
• Aaaahhh you’re right, I think I’ve been indexing incorrectly – LexyFidds Dec 4 '18 at 1:12