Question on Linear Transformations with Lines and finding Natural Matrices.

In summary, T and S are linear transformations defined by T(x; y) = (5x + y ; 2x + 2y) and S(x; y) = (3x + 2y ; x) respectively. The image of the line 2x + 3y = 5 under T is a line with equation 13x + 9y = 5. The natural matrices of the linear transformations T o S and T^-1 can be found to be [[5 1][2 2]] and [[1 -1/2][-1/2 5/2]] respectively.
  • #1
Wesc
12
1
Let T : R2 -> R2 and S : R2 -> R2 be linear transformations de fined by:
T(x; y) = (5x + y ; 2x + 2y) and S(x; y) = (3x + 2y ; x):

(i). Find the image of the line 2x + 3y = 5 under T.
(ii). Find the natural matrices of the linear transformations T o S
and T^-1

Sorry, I haven't done this topic in 3 months now, and this question came up and I'm really struggling to come up with a solution. I drew out a graph to try to visualise what the answer could be, but again nothing seemed to work out :/ I mean, I can generally do transformation of a point questions fine, but have never come across transforming a line, so any help would be much appreciated, thanks.
 
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  • #2
Nevermind, I got a solution off a friend.
 

Related to Question on Linear Transformations with Lines and finding Natural Matrices.

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space onto another in a way that preserves its linear structure. In other words, the output of a linear transformation is always a linear combination of its input.

2. How do you represent a linear transformation?

A linear transformation can be represented by a matrix. The matrix representation of a linear transformation is called its corresponding matrix, and it can be used to perform calculations and solve problems related to the transformation.

3. How do you determine if a line is transformed by a linear transformation?

A line is transformed by a linear transformation if the transformation preserves its linearity. This means that the transformed line should still be a straight line and should not become curved or distorted in any way.

4. What is a natural matrix in relation to linear transformations?

A natural matrix is a type of matrix that represents a linear transformation between two vector spaces, where the rows correspond to the basis vectors of the input space and the columns correspond to the basis vectors of the output space.

5. How can you find the natural matrix for a linear transformation?

To find the natural matrix for a linear transformation, you first need to determine the basis vectors for both the input and output spaces. Then, the natural matrix can be constructed by placing the coefficients of the basis vectors in the corresponding rows and columns of the matrix, where the coefficients represent the transformation of the basis vectors.

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