Transition and coordinate matrices

In summary: You'll get better at it if you do, and it's a good habit to get into.In summary, the conversation discusses finding the transition matrix and coordinate matrix for two bases, B and B', in R2. The transition matrix P is found to be 2/5 6/5 3/5 4/5, and the coordinate matrix [p]B is (-3, 7/2). Using P, the coordinate matrix [p]B' is found to be (3, 1). The correctness of these answers can be checked by verifying that the vectors are correctly mapped using the transition matrix.
  • #1
fattycakez
21
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Homework Statement


Consider the bases B = {b1,b2} and B' = {b'1,b'2} for R2, where

b1=(1, -1), b2=(2,0), and b'1=(1,2), b'2=(1,-3)

a. Find the transition matrix P from B to B'
b. Compute the coordinate matrix [p]B, where p=(4,3); then use the transition matrix P to compute [p]B'

Homework Equations

The Attempt at a Solution


a. I found the transition matrix P to be
2/5 6/5
3/5 4/5

b. I found [p]B to be (-3, 7/2)
and then I found [p]B' to be (3,1)

Does this look correct? I am tripping out because this seems like something we covered in class months ago but this problem was just assigned this week. Am I missing something? Would the problem be different if B' represented an orthogonal basis? Any help is appreciated :)
 
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  • #2
fattycakez said:
Does this look correct? I am tripping out because this seems like something we covered in class months ago but this problem was just assigned this week. Am I missing something? Would the problem be different if B' represented an orthogonal basis? Any help is appreciated :)

You should be able to check your answers for yourself on this one. For example, to check the vectors are correct you simply check that:

##-3b_1 + (7/2)b_2 = -3(1, -1) + 7/2(2, 0) = (-3, 3) + (7, 0) = (4,3)##

So, (4, 3) is indeed (3, 7/2) in basis B.

Etc.

And, you can use the transition matrix to operate on the vectors to check that it does map the vectors correctly. This is often a good thing to check in any case.

Everything looks correct, but I'd suggest you check them yourself.
 
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FAQ: Transition and coordinate matrices

What is a transition matrix?

A transition matrix is a square matrix that represents the probabilities of transitioning from one state to another in a Markov chain. It is usually denoted by the letter P and each element in the matrix represents the probability of transitioning from one state to another.

How is a transition matrix used in probability?

A transition matrix is used in probability to model the behavior of a system where the outcomes are dependent on the previous state. It is useful in predicting future outcomes and analyzing the long-term behavior of a system.

What is a coordinate matrix?

A coordinate matrix is a matrix that represents the coordinates of a vector in a given basis. It is used to transform a vector from one basis to another. The columns of the matrix are the coordinates of the vector in the new basis.

How are transition and coordinate matrices related?

Transition and coordinate matrices are related in the sense that they both involve matrix transformations. A transition matrix transforms a vector from one state to another, while a coordinate matrix transforms a vector from one basis to another. Both matrices are used to analyze and predict the behavior of a system or vector.

How can transition and coordinate matrices be applied in real life?

Transition and coordinate matrices have various applications in real life, such as in finance, biology, and engineering. In finance, they can be used to model stock prices and predict future trends. In biology, they can be used to analyze population growth and disease spread. In engineering, they can be used to model and predict the behavior of complex systems such as traffic flow or power grids.

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