Help me to solve this exercise.

[rumble182]

I have some trouble with 1st question.

 

[Dr.Peterson]

Please show us where you are having trouble, so we can know what help to give you. What matrix did you make? Why are you not sure of it?

Or, at least, show us an example you have been given, of a matrix associated with a mapping, so we can see what you have learned.

 

[rumble182]

Please show us where you are having trouble, so we can know what help to give you. What matrix did you make? Why are you not sure of it?

Or, at least, show us an example you have been given, of a matrix associated with a mapping, so we can see what you have learned.

I know how to solve steps b, c, d but I cannot understand how change linear map to canonical bases on (a).

 

[Romsek]

I know how to solve steps b, c, d but I cannot understand how change linear map to canonical bases on (a).

it's a lot easier than you seem to be making it.

[MATH]A \begin{pmatrix}x\\y\\z\\t\end{pmatrix} = \begin{pmatrix}y\\x\\z\\t\end{pmatrix}[/MATH]
Solve for [MATH]A[/MATH]

 

[HallsofIvy]

Do you understand what the "canonical basis" is? The canonical basis for R⁴ consists of the vectors
$$\begin{pmatrix}1 \\ 0 \\ 0 \\ 0 \end{pmatrix}$$, $$\begin{pmatrix}0 \\ 1 \\ 0 \\ 0 \end{pmatrix}$$, $$\begin{pmatrix}0 \\ 0 \\ 1 \\ 0 \end{pmatrix}$$ and $$\begin{pmatrix}0 \\ 0 \\ 0 \\ 1 \end{pmatrix}$$.

Equivalently to what Romsek suggested, since A is from R⁴ to R⁴, it can be written as a 4 by 4 matrix, $$A= \begin{pmatrix}a & b & c & d \\ e & f & g & h \\i & j & k & i \\ m & n & o & p\end{pmatrix}$$.

Now, since A maps (x, y, z. t) to (y, x, z. t). it maps (1, 0, 0, 0) to (0, 1, 0, 0):
$$\begin{pmatrix}a & b & c & d \\ e & f & g & h \\i & j & k & i \\ m & n & o & p\end{pmatrix}\begin{pmatrix}1 \\ 0 \\ 0 \\ 0 \end{pmatrix}= \begin{pmatrix} a \\ e \\ i \\ m\end{pmatrix}= \begin{pmatrix}0 \\ 1 \\ 0 \\ 0 \end{pmatrix}$$ so that a= 0, e= 1, i= 0, and m= 0.

And
$$\begin{pmatrix}a & b & c & d \\ e & f & g & h \\i & j & k & i \\ m & n & o & p\end{pmatrix}\begin{pmatrix}0 \\ 1 \\ 0 \\ 0 \end{pmatrix}= \begin{pmatrix} b \\ f\\ j \\ n\end{pmatrix}= \begin{pmatrix}1 \\ 0 \\ 0 \\ 0 \end{pmatrix}$$ so that b= 1, f= 0, j= 0, and n= 0.

 

[Steven G]

You really need to learn the matrices which performs the elementary row operations.
Every Linear Algebra students should/must know them.