Finding the Transformation Matrix for Linear Transformations in R^3

In summary: Can you please explain it in more detail?In summary, the problem lies in finding the transformation matrix for A.
  • #1
sintec
10
0
In a vector space R^3, is given a transformation A with a subscript A(x1,x2,x3)=(2*x1+x2, x1+x2+2*x3, -x2+x3).
Linear transformation B has in the basis; (1,1,1), (1,0,1), (1,-1,0) a matrix T:

[-1 2 3]
[ 1 1 0]
[ 0 1 1]
Write down a matrix which belongs to the transformation AB in the standard basis of the vector space R^3.

I just can't solve this problem. Please help!
 
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  • #2
Hmm, the standard basis is (1,0,0) (0,1,0) (0,0,1)
You will need first to find the transformation matrix for A.

Then you will need transform T, to a matrix to change from the standard basis to the standard basis. You do this by multiplying it by the matrix to change from the given basis to the standard basis, and then multiplying the result by the matrix to change from the standard basis to the given basis.

B = G * T * S

G: Matrix to change from the given basis to the standard basis
S: Matrix to change from the standard basis to the given basis

All you need to then is to multiply the 2 matrix.

If you need more complete explanition on any step, let me know.
 
  • #3
Yes, a more complete explanation would be very helpful.
Thanks, for the reply!
 
  • #4
Well, at which step does the problem lie?
 
  • #5
You will need first to find the transformation matrix for A.

How do i do that?

Then you will need transform T, to a matrix to change from the standard basis to the standard basis

From the standard basis to the standard basis?

I actually have problems with transforming a matrix from one basis to another.
 
  • #6
To find a transformation matrix:

you calculate the image of the vectors of the basis:
so for example A(e1) = (2,1,0)
and then write it as the product of the vectors of the basis you transfering to:
(2,1,0) = 2e1 + e2 + 0 e3
And last write it as a matrix, where this would fill a row, in the matrix. You will get a 3x3 Matrix, because you have the 3 basis vectors.

To find the Matrix to transform from a basis to another, you do the same, but just express the vectors of the matrix you want to transform to as vectors of the matrix you transform from.
 
  • #7
I think i got it now. So the transformation matrix A is : [2 1 0 ]
[1 1 -1]
[0 2 1]

Is that correct?

Then you will need transform T, to a matrix to change from the standard basis to the standard basis. You do this by multiplying it by the matrix to change from the given basis to the standard basis, and then multiplying the result by the matrix to change from the standard basis to the given basis. B = G * T * S G: Matrix to change from the given basis to the standard basis S: Matrix to change from the standard basis to the given basis

I'm not sure how to do that.
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another, preserving the operations of addition and scalar multiplication. In simpler terms, it is a way to transform one set of data into another set of data using a set of linear equations.

2. What is the purpose of linear transformations in science?

Linear transformations are used in many scientific fields, including physics, chemistry, and engineering. They are particularly useful in data analysis and modeling, as they allow for the transformation of data into a more useful or meaningful form. They can also be used to solve systems of linear equations and study geometric transformations.

3. How do you perform a linear transformation?

To perform a linear transformation, you first need to define the transformation matrix, which contains the coefficients of the linear equations. Then, you can multiply this matrix by the vector or set of data that you want to transform. This will result in a new vector or set of data that has been transformed by the linear equations.

4. What are some common types of linear transformations?

Some common types of linear transformations include scaling, rotation, reflection, and shearing. Scaling involves changing the size of an object or set of data. Rotation involves rotating an object or set of data around a fixed point. Reflection involves flipping an object or set of data across a line of symmetry. Shearing involves changing the shape of an object or set of data by shifting one axis relative to another.

5. How are linear transformations related to matrices?

Linear transformations can be represented using matrices, with each column or row representing a different variable or dimension. The transformation matrix contains the coefficients of the linear equations used to transform the data. This matrix multiplication allows for the efficient computation of linear transformations, making them an essential tool in data analysis and modeling.

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