Finding a change of basis matrix given S and T

Question

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I've stuck in this problem for a couple of hours. I've found examples on the Internet, but it seems to me that the problem I'm dealing with doesn't provide all the information I need. Am I right or there does exist a way to solve it?

Please help me, thanks!

Answer 1

Let $M$ be the desired matrix such that $$M = \begin{pmatrix}M_{1,1}&M_{1,2}\\M_{2,1}&M_{2,2}\end{pmatrix}$$ where $M_{1,1},M_{1,2},M_{2,1}$ and $M_{2,2}$ are constants to be determined. From the problem statement that you provided we are aware what the images of the basis vectors of $T$ are under this Linear Tranformation or (Change of Basis) consequently $$\begin{pmatrix}M_{1,1}&M_{1,2}\\M_{2,1}&M_{2,2}\end{pmatrix} \begin{pmatrix}1\\1\end{pmatrix} = \begin{pmatrix}1\\2\end{pmatrix}\tag{1}$$ and $$\begin{pmatrix}M_{1,1}&M_{1,2}\\M_{2,1}&M_{2,2}\end{pmatrix} \begin{pmatrix}2\\3\end{pmatrix} = \begin{pmatrix}0\\1\end{pmatrix}\tag{2}$$

$(1)$ implies that $$\begin{pmatrix}M_{1,1}+M_{1,2}\\M_{2,1}+M_{2,2}\end{pmatrix} = \begin{pmatrix}1\\2\end{pmatrix}\tag{3}$$

similary $(2)$ implies that $$\begin{pmatrix}2M_{1,1}+3M_{1,2}\\2M_{2,1}+3M_{2,2}\end{pmatrix} = \begin{pmatrix}0\\1\end{pmatrix}\tag{4}$$

Systems $(3)$ and $(4)$ yield the following equations $$M_{1,1}+M_{1,2} = 1\\ 2M_{1,1}+3M_{1,2} = 0$$

and $$M_{2,1}+M_{2,2} = 2\\ 2M_{2,1}+3M_{2,2} = 1$$

solving the above system of equations will yield $$M = \begin{pmatrix}3&-2\\5&-3\end{pmatrix}$$

I suggest you compute the images of the basis vectors of $T$ using $M$ just to be sure

Answer 2

Given $S=\{[1,2]^t,[0,1]^t\}$ and $T=\{[1,1]^t,[2,3]^t\}$ we want to find the change-of-basis matrix $P$ from $T$ to $S$.

Using matrices:

$[S|T]\rightarrow[I_2|P]$ by elementary row operations.

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

Then $R_2=-2R_1+R_2$

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

Where $P=$$\left[\begin{array}{cc} 1& 2\\ -1& -1\\ \end{array}\right]$.

Answer 3

In order to do a change of basis question, you take the matrix and flip it from S to T (originally it stated from T to S)in order to solve for the identity matrix. So in this case set up the matrix in the way I described above and use row reduction to get the left matrix equal to the identity matrix. Can you please set up both matrixes and show me what you got?