0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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

$\endgroup$
0
$\begingroup$

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?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.