This is an old revision of the document!
1. Let $V = \R^2$, and let $v = (3,2)$ in the Cartesian basis of $\R^2$. Now, we choose another basis as follows $$ e_1 = (2,1), \quad e_2 = (0,1) $$. Expand $v$ in terms of $e_1, e_2$.
2. Let $V = \{a + bt + ct^2 \mid a, b, c \in \R\} $ be the space of polynomials of degree at most 2. Let $f_1(t), f_2(t), f_3(t)$ be three elements in $V$, given as follows $$ f_1(2) = 1, \quad f_1(3) = 0, \quad f_1(5) = 0 $$ $$ f_2(2) = 0, \quad f_1(3) = 1, \quad f_1(5) = 0 $$ $$ f_3(2) = 0, \quad f_1(3) = 0, \quad f_1(5) = 1 $$ Then
3. Consider the new coordiante $(u,v)$ on $\R^2$, related to $(x,y)$ by $$ x = \cosh(u) \cos(v), \quad y = \sinh(u) \sin(v) $$