How do I express ordered bases for polynomials as a matrices? Linear Algebra. I am given 2 ordered bases for polynomials of degree 1 or less. I am asked to find the change of basis matrix that converts vectors with respect to basis B1 into vectors with respect to basis B. I need help with determining how to express basis B1 as a matrix.
B = {1, x} .
B1 = {x - 4, 1}.
If I'm not mistaken, basis B expressed as a matrix would be:
$$
    \begin{bmatrix}
    1 & 0 \\
    0 & 1 
    \end{bmatrix}
$$

I am having trouble with how to express basis B1 into a matrix form. Would it be:
$$
    \begin{bmatrix}
     1 & 0 \\
    -4 & 1 
    \end{bmatrix}
$$
or
$$
    \begin{bmatrix}
     -4 & 1 \\
    1 & 0
    \end{bmatrix}
$$

I tried solving this problem using $$
    \begin{bmatrix}
     1 & 0 \\
    -4 & 1 
    \end{bmatrix}
$$ The change of basis matrix I got was 
$$
    \begin{bmatrix}
     1 & 0 \\
    -4 & 1 
    \end{bmatrix}
$$
Thanks.
 A: Let be $\vec v \in \mathbb{R^2}$ a vector and consider two different basis:
$$B_v:{\vec v_1,\vec v_2}$$ and $$B_w:{\vec w_1,\vec w_2}$$
Vector $\vec v $ can be represented in two ways:
$$\vec v =a_1 \cdot \vec v_1+a_2 \cdot \vec v_2 =x_1 \cdot \vec w_1+x_2 \cdot \vec w_2$$
or in matrix form:
$$\vec v =V \cdot a=W\cdot x$$
NOTE  matrices V and W have the corresponding vectors of the basis as columns

Now suppose we know the components of $\vec v$ with respect to basis $B_v$ and we want to find the $x$ components of $\vec v$ with respect to basis $B_w$, thus:
$$V \cdot a=W\cdot x \implies x=W^{-1}Va=Ma$$
matrix $M=W^{-1}V$ is the matrix of change of basis from $B_v$ to $B_w$.

NOTE
This concept can be extended to vectors $\vec v \in \mathbb{R^n}$.

In your case, assuming $B$ as the canonical basis:
W= \begin{bmatrix}
    1 & 0 \\
    0 & 1 
    \end{bmatrix}
and 
V=\begin{bmatrix}
     -4 & 1 \\
    1 & 0
    \end{bmatrix}
thus
$$M=W^{-1}V=IV=V=\begin{bmatrix}
     -4 & 1 \\
    1 & 0
    \end{bmatrix}$$

A: If $V$ is a a vector space with ordered basis $B$ and $B_1$, then the change of basis matrix (from $B_1$ to $B$) is the matrix of linear transformation of the identity map denoted by $[I]_{B_1}^{B}$.
In particular when, $V=P_1(\Bbb R)$ and $B,B_1$ are as above then,

$I:V \to V $ then the change of basis from $B_1$ to $B$ is obtained as,
$I(x-4)=x-4=-4(1)+1(x)$ and $I(1)=1=1(1)+0(x)$
$[I]_{B_1}^{B}=
\begin{bmatrix}
     -4 & 1 \\
    1 & 0 
\end{bmatrix}
$

