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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.