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