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

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Homework Help Overview

The discussion revolves around finding the transition matrix from one basis, denoted as 'a', to another basis 'r'. The bases are defined in terms of linear combinations of vectors, and the original poster expresses confusion regarding the process of constructing the transition matrix.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to express the basis 'r' in terms of basis 'a' and seeks clarification on how to represent this in matrix form. Some participants suggest writing the equations in matrix form, while others question the understanding of matrix multiplication and coefficients.

Discussion Status

Participants are exploring different interpretations of the problem, with some providing guidance on how to set up the equations in matrix form. The original poster is actively seeking clarification on the coefficients involved and the process of obtaining the transition matrix.

Contextual Notes

There is an indication that the original poster may lack some foundational knowledge in vectors and matrices, which is affecting their ability to tackle the problem effectively. This has led to suggestions for additional resources to aid understanding.

concon
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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?
 
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concon said:

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{* &amp; * &amp; * &amp; * \\ * &amp; * &amp; * &amp; * \\ * &amp; * &amp; * &amp; * \\ * &amp; * &amp; * &amp; * } <br /> \pmatrix{a_1 \\a_2 \\a_3 \\a_4}
 
Last edited:
Ray Vickson said:
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{* &amp; * &amp; * &amp; * \\ * &amp; * &amp; * &amp; * \\ * &amp; * &amp; * &amp; * \\ * &amp; * &amp; * &amp; * } <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
 
concon said:
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.
 
Ray Vickson said:
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?
 
Ray Vickson said:
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?
 

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