You are given that T is a linear transformation from vector space U to vector space V and want to prove that the "range of T", R, the set of vectors, v, in V such that v= Tu for some u in U, is a subspace of V. Okay, it is clear from the definition of range of T that R is a subset of V. To show it is a subspace you only need to that it is closed under vector addition and scalar multiplication.
That is, if \(\displaystyle v_1\) and \(\displaystyle v_2\) are vectors in R and k is a scalar, then \(\displaystyle v_1+ v_2\) and \(\displaystyle kv_1\) are also in R. If \(\displaystyle v_1\) and \(\displaystyle v_2\) are in R there exist \(\displaystyle u_1\) and \(\displaystyle u_2\) such that \(\displaystyle v_1= Tu_1\) and \(\displaystyle v_2= Tu_2\). Now what can you say about \(\displaystyle v_1+ v_2\) and \(\displaystyle kv_1\)?
