1. The problem statement, all variables and given/known data This is from Serge Lang's "Linear Algebra, 3rd Edition", page 15. Consider the vector space of all functions of a variable t. Show that the following pairs of functions are linearly independent: (a) 1,t (b) t, t2 (c) t, 4 2. Relevant equations 3. The attempt at a solution I understand how to DO the problems and attain the correct results, but I don't understand WHY it works. Looking for some insight please. For example, for part (b) my answer would be to set up an equation with two numbers a and b: at + bt2=0. I would first set t = 1 which shows a+b=0. Then I would set t =-1, showing a=b, therefore a=b=0, showing the two functions cannot be written as linear combinations of one another. Thanks in advance. Trying to learn this on my own so don't have a teacher to reach out to.