Notations: L(V,W) stands for a linear transformation vector space form vector space V to W. rk(?) stands for the rank of "?". Question: Let τ,σ ∈L(V,W) , show that rk(τ + σ) ≤ rk(τ) + rk(σ). I want to know wether the way I'm thinking is right or not, or there's a better explanation. My thought is: Since every linear tansformation is reprensted by a matrix, also is τ + σ. rk(τ) and rk(σ) roughly means the numbers of basis vectors of their own matrix column spaces, so the combination τ + σ only preserves the distinct basis vectors from the column spaces of the matirces of τ and σ, namely, rk(τ + σ) ≤ rk(τ) + rk(σ) holds. Thanks for any help.