Given V_R = R^2, then let V_R have the standard basis B=B'={(1,0), (0,1)} and let L_theta((1,0)) = (cos theta, sin theta) , L_theta((0,1)) = (-sin theta, cos theta).

I understand how the bases work, but how would you go about obtaining the new vector by way of this linear transformation? For example, how you would go about computing L_(pi/4) (2,1)?

I'm thinking you would need to multiply the first L_theta by two since the first coordinate in the input is 2, but how do you "combine" L_theta((1,0)) and L_theta ((0,1)) to obtain your final answer?

Or, in lieu of giving an answer or hint, can someone at least tell me what I can search under to find more information on this? I'm not sure what to call it, and Googling "linear transformations" only gives theoretical stuff. I haven't been able to find anything like this.