Inner product and unit vector $u_1 = (1, -1)'$ and $u_2 = (1, 1)'$ are two vector of $R^2$. Endow $R^2$ with an inner product such that $||u_1|| = 1$ and $||u_2|| = 1$. Well, honestly, I don't completely understand what the problem asks. Endow $R^2$ with inner product? Then I tried the inner product of $u_1$ $u_2$. So, $<u_1,u_2> $ $= (1,-1)*(1,1)=0$. Then two vectors are orthogonal. But I don't know to how to proceed. Do i have show that $||u_1|| = 1 $ and $||u_2|| = 1 $? if so, what theorem or formula should I use? Thanks in advance!

An inner product is a function of two vectors into $\Bbb R$ that satisfies certain properties. You are asked to find a function $f((a,b),(c,d))$ that satisfies these. You can't have $u_1=1$ because $u_1$ and $1$ are different kinds of things. You can have $f(u_1,u_1)=1$, which is what you want. The required linearity is a powerful constraint. Express any two vectors in the basis of $u_1,u_2$ and you know their inner product.