Help Solving Linear Transformation Problem in R^2

In summary, to solve your problem with linear transformations and transfer matrices, you need to find the transfer matrices from B' to the canonical basis of R^2, from the canonical basis of P to the B basis, and then use these transfer matrices to find the matrix L in the canonical basis of P and R^2.
  • #1
akimbo
1
0
Hey i got a problem here but still without correction so if you guys can help me , thanks in advance I'm stuck there

We have L : P -> R^2
L is a linear transformation with :

[tex]B = \left\{1-x^{2},2x,1+2x+3x^{2} \right\} \; and \; B' = \begin{Bmatrix} \begin{bmatrix} 1\\-1 \end{bmatrix} \begin{bmatrix} 2\\0 \end{bmatrix} \end{Bmatrix} as \; [L]^{B'}_{B} = \begin{bmatrix} 2 &-1 &3 \\ 3&1 & 0 \end{bmatrix}[/tex]

I have to find
1/ the transfer matrix from B' to the canonical basis of R^2
also
2/ the transfer matrix from the canonical basis ( 1 , x , x2 )of P in the B basis
3/ find matrix L in the canonical basis of P and R^2
 
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  • #2

Thank you for reaching out for help. It seems like you are working on linear transformations and transfer matrices. Let me try to explain the steps you need to take to answer the questions you have.

1. To find the transfer matrix from B' to the canonical basis of R^2, you need to first find the inverse of the matrix [L]^{B'}_{B}. This can be done by using the Gauss-Jordan elimination method or by using the inverse matrix formula. Once you have the inverse, you can simply multiply it with the matrix [L]^{B'}_{B}. The resulting matrix will be the transfer matrix from B' to the canonical basis of R^2.

2. To find the transfer matrix from the canonical basis of P (1, x, x^2) to the B basis, you need to first find the matrix ^{B}_{P} which represents the identity transformation from P to B. This can be done by writing the vectors in B as linear combinations of the vectors in the canonical basis of P. Then, you can simply multiply ^{B}_{P} with the matrix [L]^{B}_{B'} to get the transfer matrix from the canonical basis of P to the B basis.

3. To find the matrix L in the canonical basis of P and R^2, you can use the transfer matrices you have found in the previous steps. The matrix L in the canonical basis of P and R^2 can be written as [L]^{B}_{P} [L]^{B'}_{B} ^{B}_{P}.

I hope this helps you solve the problem. Good luck!
 

FAQ: Help Solving Linear Transformation Problem in R^2

What is a linear transformation in R^2?

A linear transformation in R^2 is a mathematical function that maps points from a two-dimensional coordinate system to another two-dimensional coordinate system while preserving the straightness of lines.

How do I solve a linear transformation problem in R^2?

To solve a linear transformation problem in R^2, you need to follow these steps:

  1. Identify the transformation matrix, which represents the transformation in terms of scaling, rotation, and shearing.
  2. Apply the transformation matrix to each point in the original coordinate system to get the corresponding points in the new coordinate system.
  3. Plot the original points and the transformed points on a graph to visualize the transformation.

What are some common applications of linear transformations in R^2?

Linear transformations in R^2 have various applications in fields such as computer graphics, engineering, physics, and economics. They are used to model and analyze linear systems, create geometric transformations in computer graphics, and solve optimization problems in economics.

How do I check if a transformation in R^2 is linear?

To check if a transformation in R^2 is linear, you can use the properties of linearity, such as the preservation of addition and scalar multiplication. If the transformation satisfies these properties, then it is linear.

Can a linear transformation in R^2 have a determinant of 0?

No, a linear transformation in R^2 cannot have a determinant of 0. The determinant of a transformation matrix represents the scaling factor of the transformation, and a determinant of 0 would mean that the transformation does not preserve the area of shapes, which goes against the definition of a linear transformation.

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