A question in basis transformation

In summary, the conversation discusses the process of building the T transformation using the standard basis and the B basis. It also addresses an error in forming the "S-1" transformation matrix and the confusion over the placement of vectors in matrices. The correct method is to use the basis vectors as columns in the matrix.
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  • #2
You could check it your self, In terms of the standard basis, Ttr(1, 1, 1)= (3, 3, 7), Ttr(1, 0, 0)= (2, 1, 1), and Ttr(0, 0, 1)= (0, 0, 5). In terms of the B basis, those results would be (3, 3, 7)= a(1, 1, 1,)+ b(1, 0, 0)= c(0, 0, 1)= (a+ b, a, a+ c) so we have a+ b= 3, a= 3, a+ c= 7 which gives a= 3, b= 0, c= 4 or <3, 0, 4> (I am using "< >" for vectors written in the B basis). Similarly, (2, 1, 1) gives a= 1, b= 1, c= 0 or <1, 1, 0> and (0, 0, 5) gives < 0, 0, 5>. If you try to do everything in the "B" basis: multiply your "TB" matrix by <1, 0, 0>, <0, 1, 0>, and <0, 0, 1> you get the first, second, and third columns, respectively. And they are NOT the same.

You error was when you formed the "S-1" transformation matrix: you used the B basis vectors as rows and they should be columns. Use
[tex]\left(\begin{array}{ccc} 1 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 1\end{array}\right)[/tex]
instead and you should be alright.
 
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  • #3
i am confused about the vector apeareance
i what form should i put it in the matrix
as a row

i what form should i put it in the matrix
as a column?

also i was tald that in transformation we put the vectors as rows

for example:
in this sort of question i was to find the basis of V

http://img301.imageshack.us/my.php?image=img83241re6.jpg

first i thought that when a vector is signed as (x,y,z)
we flip him verticaly
and when its
(x)
(y)
(z)
then it should flip it horisontaly
but apparently that's not how it works

how it works??
how do i write the given vectors in the metrix
and in what form and in what cases??
 
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1. What is basis transformation in scientific terms?

Basis transformation is a mathematical process used in linear algebra to change the coordinate system of a vector or set of vectors. It involves finding a new set of basis vectors that can represent the same vector space in a different way.

2. Why is basis transformation important in scientific research?

Basis transformation is important because it allows scientists to analyze data or equations in different coordinate systems, which can sometimes make problem-solving easier or reveal new insights. It also has applications in areas such as computer graphics, physics, and engineering.

3. How is basis transformation different from other mathematical transformations?

Basis transformation is different from other transformations because it involves changing the coordinate system itself, rather than just transforming individual points or vectors within a coordinate system. It also maintains the same vector space, whereas other transformations may change the vector space.

4. Can you give an example of a basis transformation in a scientific context?

One example of a basis transformation in a scientific context is the transformation from Cartesian coordinates to cylindrical coordinates. This is commonly used in physics and engineering to describe the motion of objects in a circular or cylindrical path, such as a rotating wheel or a flying object.

5. Are there any limitations to basis transformation?

Yes, there are limitations to basis transformation. Some vector spaces may not have a valid basis, making transformation impossible. Additionally, some transformations may result in a more complex or difficult to interpret coordinate system, making it less useful for analysis. It is important for scientists to carefully consider the implications of basis transformation before applying it to their research.

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