Originally Posted by

**hiddy** Hello

right now Im studying for my linear algebra exam and I think I pretty much understood the basic concepts but then I tried the following and I am not able to solve it, can someone help me with this:

In the vector space $\displaystyle \mathbb{R}^\mathbb{R}$ is U a subspace with the following polynomials as basis $\displaystyle \mathbf{B}$:

$\displaystyle f_0:x \mapsto 1$

$\displaystyle f_1:x \mapsto x$

$\displaystyle f_2:x \mapsto x^2$

$\displaystyle f_3:x \mapsto x^3$

Additionally there are the following functions:

$\displaystyle p_:0x \mapsto x^3$

$\displaystyle p_1:x \mapsto x^2(1-x)$

$\displaystyle p_2:x \mapsto x(1-x)^2$

$\displaystyle p_3:x \mapsto (1-x)^3$

which form a family $\displaystyle \mathbf{C}:=(p_0,p_1,p_2,p_3)$

a) Calculate the Transformation Matrix $\displaystyle <\mathbf{B}^\ast,\mathbf{C}>$ and the inverse.

b) Show why $\displaystyle \mathbf{C}$ is a basis

c) Calculate the Coordinates $\displaystyle <\mathbf{C},q> $ of $\displaystyle q: x \mapsto -x^3+3x^2-4x+1$

d) For the linear map $\displaystyle g_2: f \mapsto f(2) $calculate the Transformation matrices $\displaystyle <g_2,\mathbf{B}>$ and $\displaystyle <g_2,\mathbf{C}>$.

I have no clue how to begin. I think i know how to solve b) and inverting a matrix is not a problem but the rest is not clear to me!

Can you help me, please!