How can I find the transition matrix from bases a to r?

[concon]

Homework Statement

let a= {a1, a2, a3, a4}
and b={b1,...,b4}
and r = {r1,...,r4}

Also,
b1 = 4a1
b2 = 8a1 + 7a2
b3 = 4a1 + 4a2 + 4a3
b4 = 9a1 + 5a2 + 8a3 + 5a4

and

r1 = 3b4
r2= 4b3 + 6b4
r3 = 9b2 + 3b3 + 9b4
r4 = 6b1 + 5b2 + 3b3 + 5b4

Find transition matrix from basis a to r.

Homework Equations

To find transition matrix I know you make matrix with on one side one basis and the other side the other basis and turn one of them into the identity matrix.
Hard to visually represent this equation, but hopefully you know what I mean.

The Attempt at a Solution

Normally these problems are really easy in class with bases like {(1,2), (2,1)}
I have no idea how to solve this one.
thus far I have but r in terms of a:

r1 = 27a1 + 15a2 + 24a3 + 15a4
r2 = 70a1 + 46a2 + 64a3 + 30a4
r3 = 165a1 + 120a2 + 84a3 + 45a4
r4 = 121a1 + 72a2 + 40a3 + 25a4

What do I do from here?

 

[Ray Vickson]

Homework Statement

let a= {a1, a2, a3, a4}
and b={b1,...,b4}
and r = {r1,...,r4}

Also,
b1 = 4a1
b2 = 8a1 + 7a2
b3 = 4a1 + 4a2 + 4a3
b4 = 9a1 + 5a2 + 8a3 + 5a4

and

r1 = 3b4
r2= 4b3 + 6b4
r3 = 9b2 + 3b3 + 9b4
r4 = 6b1 + 5b2 + 3b3 + 5b4

Find transition matrix from basis a to r.

Homework Equations

To find transition matrix I know you make matrix with on one side one basis and the other side the other basis and turn one of them into the identity matrix.
Hard to visually represent this equation, but hopefully you know what I mean.

The Attempt at a Solution

Normally these problems are really easy in class with bases like {(1,2), (2,1)}
I have no idea how to solve this one.
thus far I have but r in terms of a:

r1 = 27a1 + 15a2 + 24a3 + 15a4
r2 = 70a1 + 46a2 + 64a3 + 30a4
r3 = 165a1 + 120a2 + 84a3 + 45a4
r4 = 121a1 + 72a2 + 40a3 + 25a4

What do I do from here?

You've done it; now just write these equations in matrix form. That is, fill in the blanks in
$$\pmatrix{r_1\\r_2\\r_3\\r_4} =<br /> \pmatrix{* & * & * & * \\ * & * & * & * \\ * & * & * & * \\ * & * & * & * } <br /> \pmatrix{a_1 \\a_2 \\a_3 \\a_4}$$

 

[concon]

You've done it; now just write these equations in matrix form. That is, fill in the blanks in
$$\pmatrix{r_1\\r_2\\r_3\\r_4} =<br /> \pmatrix{* & * & * & * \\ * & * & * & * \\ * & * & * & * \\ * & * & * & * } <br /> \pmatrix{a_1 \\a_2 \\a_3 \\a_4}$$

Okay, I still don't understand. Do I put the coefficients of r and a and solve for whatever is in the matrix of *?

What are the coefficients of a? All ones? I'm confused

 

[Ray Vickson]

Okay, I still don't understand. Do I put the coefficients of r and a and solve for whatever is in the matrix of *?

What are the coefficients of a? All ones? I'm confused

It sounds like you are saying that you do not understand vectors, matrices and matrix multiplication. You need to go to the library and take and read out a book on the subject; it is much too lengthy to be presented in a help forum. You can also try on-line notes and tutorials on the subject.

Please note: I am NOT trying to insult you; it really does sound to me as though you do not have the background to tackle the question.

 

[concon]

It sounds like you are saying that you do not understand vectors, matrices and matrix multiplication. You need to go to the library and take and read out a book on the subject; it is much too lengthy to be presented in a help forum. You can also try on-line notes and tutorials on the subject.

Please note: I am NOT trying to insult you; it really does sound to me as though you do not have the background to tackle the question.

no i get that the matrix with the astrisk's should contain the coefficients of r. What are the components of the matrix with a1 through a4?

 

[concon]

It sounds like you are saying that you do not understand vectors, matrices and matrix multiplication. You need to go to the library and take and read out a book on the subject; it is much too lengthy to be presented in a help forum. You can also try on-line notes and tutorials on the subject.

Please note: I am NOT trying to insult you; it really does sound to me as though you do not have the background to tackle the question.

Wait hold on I think I just realized how to solve this problem. You can get one transition matrix from another transition matrix by taking the inverse right? So can I just take inverse of the matrix with the asterisks?