A question about the rank of the sum of linear transformations

[sanctifier]

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.

 

[adriank]

Show that the im(τ + σ) is a subspace of im(τ) + im(σ). Remember that in general, rk(τ) = dim(im(τ)).

(By im, I mean image.)

 

[sanctifier]

Show that the im(τ + σ) is a subspace of im(τ) + im(σ). Remember that in general, rk(τ) = dim(im(τ)).

(By im, I mean image.)

Indeed, it's a simpler way to solve the question.

Thanks!