Came across this problem during my studies... Here it is:

be the functions defined by and where r is not equal to s. Prove that and are linearly independent in .

However, I'm confused about what I can allow to be. Because if then and are both equal to 1.

Thus, I can form the equation

If then letting and , the equation is true. Thus, aren't the equations linearly dependent?

EDIT: Forgot to include another problem I'm having trouble with!

Let u, v, and w be distinct vectors of a vector space V. Show that if {u, v, w} is a basis for V, then {u+v+w, v+w, w} is also a basis for V.

I have no idea how to even start that one!