Coordinates

1. Let ${\cal B} = \{\alpha_1,\alpha_2,\alpha_3\}$ be an ordered basis for $\mathbb{R}^3$ where
   $$\alpha_1 = (1,0,-1), \;\;\;\alpha_2 = (1,1,1), \;\;\; \alpha_3 = (1,0,0)$$
   What are the coordinates of $(a,b,c)$ in the ordered basis ${\cal B}$?
2. Let $\alpha = (x_1,x_2)$ and $\beta = (y_1,y_2)$ be vectors in $\mathbb{R}^2$ such that
   $$x_1y_1 + x_2y_2 = 0$$    and $$x_1^2 + x_2^2 = y_1^2 + y^2 = 1$$
   Show that ${\cal B} = \{\alpha,\beta\}$ is a basis for $\mathbb{R}^2$. Find the coordinates of $(a,b)$ in this ordered basis. What do the conditions mean geometrically?
3. Consider the matrix
   $${\bf P} = \left[ \begin{array}{cc} \cos{\theta} & - \sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{array}\right]$$
   Show that ${\bf P}$ is invertible. Conclude that ${\bf P}$ represents a transformation of coordinates in $\mathbb{R}^2$. What is the geometric interpretation of the change of coordinates represented by ${\bf P}$?
4. Let $V$ be a vector space over the complex numbers of all functions from $\mathbb{R}$ to ${\tt C }$, i.e., the space of all complex valued functions on the real line. Let $f_1(x) = 1$, $f_2(x) = e^{ix}$ and $f_3(x) = e^{-ix}$.
   1. Prove that $f_1,f_2,f_3$ are linearly independent.
   2. Let $g_1(x) = 1$, $g_2(x) = \cos{x}$ and $g_3(x) = \sin{x}$. Find an invertible $3 \times 3$ matrix ${\bf P}$ such that
      $$g_j = \sum_{i=1}^3 P_{ij}f_i$$

5. Let $V$ be the real vector space of all polynomial functions from $\mathbb{R}$ into $\mathbb{R}$ of degree 2 or less. Let $t$ be a fixed number and define
   $$g_1{x} = 1, \;\;\;g_2(x) = x+t,\;\;\;g_3(x)-(x+t)^2$$
   Prove that ${\cal B} = \{g_1,g_2,g_3\}$ is a basis for $V$. If
   $$f(x) = c_0 + c_1 x + c_2 x^2$$
   what are the coordinates of $f$ in this ordered basis ${\cal B}$?