Question about Linear Transformation.

Question

Suppose $S\in\mathcal{L}(V)$ with $dim\ V = n$. Let $(v_1,...,v_n)$ be basis of $V$. Define $S$ by $$S(a_1v_1+\cdots+a_nv_n)=a_1v_1.$$

My question is, is it necessary that $Sv_2=0$?

Answer 1

Yes. In particular, we can apply your definition to find $$ Sv_2 = S(0v_1 + 1v_2 + 0v_3 + \cdots + 0v_n) = 0v_1 = 0 $$

Answer 2

By properties of Linear transformation we have that

$$S(a_1v_1+\cdots+a_nv_n) = S(a_1v_1) + S(a_2v_2) +\cdots+ S(a_nv_n) = $$ $$ a_1S(v_1) + \cdots + a_nS(v_n) = a_1v_1$$

and $S$ carries Linearly independent vectors to linearly independent vectors in the transformed space. If we assume is not zero we can get a contradiction easily by using the linear independence of the transformed vectors and conclude that $S(v_2)$ must be zero.