Finding a change of Basis Matrix Of Polynomial Degree Less than 1

Question

I am doing some practice questions and I'm not too sure where to start on this one:

1. Consider the bases $B = \{p_1,p_2\}$ and $B' = \{q_1,q_2\}$ of P$_1$, the polynomials of degree $\le$ 1, where $p_1 = 2$; $p_2 = x - 1$; $q_1 = x + 3$; $q_2 = x + 1.$
   (a) Find the transition matrix from B' to B.
   (b) Find the transition matrix from B to B'.
   (c) Let $p = 2x - 1$. Compute $[p]_B$, the coordinates of $p$ with respect to $B$, and then use (b) to find $[p]_{B'}$, the coordinates of $p$ with respect to $B'$.

Any help is appreciated.

Answer 1

The transition matrix from basis $B_1$ to basis $B_2$ is found by just writing the vectors in $B_1$ in the coordinates of $B_2$. In you case, you need to find $\begin{bmatrix} p_1 \end{bmatrix}_{B'}, \begin{bmatrix} p_2 \end{bmatrix}_{B'}, \begin{bmatrix} q_1 \end{bmatrix}_{B}, \begin{bmatrix} q_2 \end{bmatrix}_{B}$. These vectors are \begin{align} p_1 &= 1 (x + 3) - 1 (x + 1), \quad \text{which implies} \\ \begin{bmatrix} p_1 \end{bmatrix}_{B'} &= \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \\ p_2 &= -1 (x + 3) + 2(x + 1), \quad \text{which implies} \\ \begin{bmatrix} p_2 \end{bmatrix}_{B'} &= \begin{bmatrix}-1 \\2 \end{bmatrix}. \end{align} Hence, the transition matrix from $B$ to $B'$ is just $$\begin{bmatrix} \begin{bmatrix} p_1 \end{bmatrix}_{B'} & \begin{bmatrix} p_1 \end{bmatrix}_{B'} \end{bmatrix} = \begin{bmatrix} 1 & -1 \\ -1 & 2 \end{bmatrix}.$$

Can you follow these steps to find the transition matrix from $B'$ to $B$? The actual computation boils down to finding the coefficients (coordinates). How did I do it?

Answer 2

There is an algorithm for finding the transition matrix and it is as following:

• Augment the matrix representing the basis for the coordinate system you want to convert to, with the matrix representing the basis for the coordinate system you want to convert from; so in this case $$ \left[ \begin{array}{c|c} B&B'\\ \end{array} \right] $$

• RREF the augmented matrix and what's on the left side of the column should be $I_n$ and on the right side it should be $T_{B'}^B$

  1. $$ \left[ \begin{array}{cc|cc} 2&-1&3&1\\ 0&1&1&1 \end{array} \right] $$

  2. $$ \left[ \begin{array}{cc|cc} 1&0&2&1\\ 0&1&1&1 \end{array} \right] $$

  3. $T_{B'}^B$ = $$ \begin{bmatrix} 2&1\\ 1&1\\ \end{bmatrix} $$