Linear Algebra - Change of Bases

In summary, The conversation is about finding the representing matrix T = K_{B_{2},B_{1}} \in \Re^{2\times2} for the change from B1 coordinates to B2 coordinates. The method involves finding K_{S,B_1} and K_{S,B_2} and using the formula K_{B_2,B_1} = K_{B_2,S} K_{S,B_1} = K_{S,B_2}^{-1}K_{S,B_1} to calculate the desired matrix.
  • #1
Mumba
27
0
Hi, again another problem:

Let B1 = {( [tex] \stackrel{1}{3}[/tex]),([tex] \stackrel{1}{2}[/tex])} and


[tex]B_{2} = [ \frac{1}{\sqrt{2}}( \stackrel{1}{1}), \frac{1}{\sqrt{2}} (\stackrel{-1}{1}) ] [/tex]


Determine the representing matrix [tex]T = K_{B_{2},B_{1}} \in \Re^{2\times2}[/tex] for the change from B1 coordinates to B2 coordinates.

I have no idea what i should do here. I ve found how to calculate the representing matrix from a domain to a codomain.
Is this the same way? Can you give me atleast a hint, please ^^.

Thx Mumba

PS. Sorry, it looks really strange. I don't know how to formate this better.
 
Physics news on Phys.org
  • #2
This reminds me of quantum mechanics 1, a course that crushed m average and of which I remember very little.

I tried a couple things and the answer i got was 1/sqrt(2) ( 2 5 )
( 0 1 )
for the matrix T. But as i said again, that's just ag uess
 
  • #3
If S denotes the standard basis for [itex]\mathbb{R}^2[/itex], do you know how to find

[tex]K_{S,B_1}[/tex] and [tex]K_{S,B_2}[/tex]?

If so, then observe that

[tex]K_{B_2,B_1} = K_{B_2,S} K_{S,B_1} = K_{S,B_2}^{-1}K_{S,B_1}[/tex]
 
  • #4
No, i don't even know what K is supposed to be...
 
  • #5
Mumba said:
No, i don't even know what K is supposed to be...

I am using your notation.

[tex]K_{S,B_1}[/tex]

is the change-of-basis matrix that transforms [itex]B_1[/itex] coordinates to [itex]S[/itex] coordinates. I suggested doing it this way because you already know how to express [itex]B_1[/itex] and [itex]B_2[/itex] in [itex]S[/itex] coordinates: that is what you are given.
 
  • #6
So i should calculate the change of base matrix for B1 to S and the inverse of the change of base matrix for B2 to S coordinates?
And multiply this to get my result?

Ok i ll try to find out how to calculate the change of base matrix ^^
Thx
 
  • #7
Is [text]K_{S,B1}[/tex] not the same as B1?
And the same for B2? So i calculate the inverste and multply them and then I am finished?
 
  • #8
the i get as matrix:

4/sqrt(2) 3/sqrt(2)
2/sqrt(2) 1/sqrt(2)

correct?
 
  • #9
Mumba said:
Is [text]K_{S,B1}[/tex] not the same as B1?
And the same for B2? So i calculate the inverste and multply them and then I am finished?

Well, [itex]B_1[/itex] is a basis (set of vectors), not a matrix, so what you said is not exactly correct. However, I think what you are trying to say is this:

[tex]K_{S,B_1}[/tex] is the matrix whose columns are the basis vectors from [itex]B_1[/itex] expressed in [itex]S[/itex] coordinates, namely

[tex]K_{S,B_1} = \left[\begin{array}{cc}1 & 1 \\ 3 & 2\end{array}\right][/tex]

and similarly for [tex]K_{S,B_2}[/tex].

So yes, now you can find [tex]K_{B_2,B_1}[/tex] as I described earlier.
 
  • #10
Coool thanks a lot, easy this way.
:D
 
  • #11
Mumba said:
the i get as matrix:

4/sqrt(2) 3/sqrt(2)
2/sqrt(2) 1/sqrt(2)

correct?

I get the same answer.
 
  • #12
Cool :)
Thx jbunniii
 

What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations through vector spaces and matrices. It is used to solve problems involving systems of linear equations, transformations, and geometric objects.

What is a change of basis in linear algebra?

A change of basis refers to the process of representing a vector or a matrix in a different coordinate system. This is achieved by finding a new set of basis vectors that can be used to express the original vector or matrix in a different way.

Why is change of basis important in linear algebra?

Change of basis is important in linear algebra because it allows us to simplify and solve complex problems involving linear transformations and systems of equations. It also helps in understanding the geometric interpretations of linear algebra concepts.

What is the process of changing bases in linear algebra?

The process of changing bases involves finding a matrix that transforms the original basis vectors into the new basis vectors. This matrix is known as the change of basis matrix and it is used to convert coordinates from one basis to another.

What are the applications of change of basis in real life?

Change of basis has many real-life applications, such as in computer graphics, data compression, and image processing. It is also used in engineering and physics to solve problems involving systems of linear equations and transformations.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
704
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
905
  • Calculus and Beyond Homework Help
Replies
6
Views
3K
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
13
Views
1K
Back
Top