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collection of linear transformations T:V to W is a vector space.

I know that the main idea is to show T1:V to W and T1:V to W, then T1+T2 is a linear transformation.

However, there is one thing that confuses me a lot. If I wanna prove V is a vector space, I need to find v, w and so on belongs to V, and then check properies one by one. For example, v+w=w+v. In this case V does not change or put in the proof process at all.

For the collection of linear transformations, I do not know how can I prove it. It seems that I have to use T1+T2(v)

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If I wanna prove V is a vector space, I need to find v, w and so on belongs to V, and then check properies one by one.

This is exactly the idea. However, you are not supposed to do this for the $V$ that appears in your problem. Your problem is asking you to verify that the collection of linear transformations $T:V\to W$ (with appropriate operations) is a vector space. Let us call this collection "$L$."

Using your words, we need to show that elements of this collection $L$ satisfy the definitions of a vector space. Let's take two elements $L$, and call them $T_1$ and $T_2$. Both are linear transformations $V \to W$. Verifying closure under linearity is basically showing that $T_1 + T_2$ is also in $L$, or in other words, $T_1+T_2$ is also a linear transformation $V \to W$, which is the "main idea" that you stated.

There is a question of what the addition in "$T_1+T_2$" means. This is your last sentence: by definition $T_1+T_2$ is the map defined by $(T_1+T_2)(x) = T_1(x)+T_2(x)$. All that is left to verify is that this map is also linear.

Response to comment: \begin{align}(T_1+T_2)(av_1+bv_2) &= T_1(av_1+bv_2) + T_2(av_1+bv_2) \\&= aT_1(v_1) + bT_1(v_2) + aT_2(v_1)+bT_2(v_2)\\&=a(T_1+T_2)(v_1) + b(T_1+T_2)(v_2).\end{align}

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    Great! Thank you for the reply. – 2017-02-06
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    I think for the addition under closure properies, I need to show (T1+T2)(av1+bv2)=a(T1+T2)v1+b(T1+T2)v2. Because that is how to prove it is a linear transformation. However, I do not know how to go next for the scalar multiplication under closure properies. Should I prove that a(T1+T2)(v) is also a linear transformation? – 2017-02-06
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    @HemingLi See my edit – 2017-02-06
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    Thank you sir. I think the edit is about the proof of addition closure property, I am wondering if you could show the scalar multiplication closure property. Sorry for any inconvenience. – 2017-02-06
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    @HemingLi Sorry I misread your comment. Yes, that follows if you prove that in general $aT$ is linear if $T$ is linear. – 2017-02-06