Linear Transformation Matrix: Inverse, Areas & Orientation Analysis

In summary, the linear transformation represented by the matrix M = (-3,2; 0,-2) results in a change in orientation from the x-axis to the y-axis and decreases the area of a square with corners at (0,0) and (1,1). To find the inverse of this transformation, the determinant of the matrix can be calculated and the inverse matrix can be found by switching the values on the main diagonal and changing the signs of the off-diagonal values.
  • #1
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Homework Statement



let f be the linear transformation represented by the matrix

M = ( -3, 2)
( 0, -2)

state what effect f has on areas, and whether f changes orientation.

Find the matrix that represents the inverse of f.



Homework Equations



N/A

The Attempt at a Solution



I think I'm over complicating this. I have drawn out the matrix on set of axes. I don't really understand the question, any help or pointers in the right direction would be greatly appreciated.
 
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  • #2
Since you have already calculated what M does to your axes (basis vectors), try figuring out what happens to a little square with corners on (0, 0) and (1, 1).

What happens to its area, for example? (At this point you may want to calculate the determinant of the matrix).

If you go from the x-axis to the y-axis you turn counterclockwise. Does the same hold for the transformed rectangle?
 

FAQ: Linear Transformation Matrix: Inverse, Areas & Orientation Analysis

What is a linear transformation matrix?

A linear transformation matrix is a mathematical representation of a linear transformation, which is a function that maps vectors from one vector space to another. It is a square matrix that contains coefficients that define how the transformation affects the input vectors.

What is the inverse of a linear transformation matrix?

The inverse of a linear transformation matrix is another matrix that, when multiplied with the original matrix, results in the identity matrix. It essentially undoes the transformation and allows for the original input vectors to be retrieved from the transformed vectors.

How is the inverse of a linear transformation matrix calculated?

The inverse of a linear transformation matrix can be calculated using various methods, such as Gaussian elimination, LU decomposition, or the Gauss-Jordan method. These methods involve manipulating the coefficients of the matrix to transform it into the identity matrix.

How can a linear transformation matrix be used for area and orientation analysis?

A linear transformation matrix can be used for area and orientation analysis by applying it to a shape or object. The transformed shape will have a different area and orientation compared to the original shape, and this information can be used to analyze the effects of the transformation.

Can a linear transformation matrix change the orientation of a shape?

Yes, a linear transformation matrix can change the orientation of a shape. This is because the matrix contains coefficients that define how the transformation affects the input vectors, including their orientation. For example, a rotation matrix can be used to rotate a shape by a certain angle, changing its orientation.

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