Matrix Transformation - A plane mapping onto itself.

In summary, the matrix M is used to find which planes map onto themselves by using its inverse to directly substitute x, y, and z into the equation of a plane. The resulting coefficients of x, y, and z should match the original coefficients in order for the plane to map onto itself. In this case, the conditions for a plane to map onto itself under the matrix M are a=c and 2a=b.
  • #1
binbagsss
1,261
11
Find which planes map onto themselves under the matrx M.

M=
1 2 0
0 1 -1
0 2 1

(in enclosed brackets - apologies for the format.).

Attempt:

Consider a plane ax+by+cz=d [1].

M^-1 :
3/3 -2/3 -2/3
0 1/3 1/3
0 -2/3 1/3

(in enclosed bracket).

- use of the inverse so that x,y,z can then be directly subbed into [1].

x= X-2/3(Y)-2/3(Z)
y=(Y)/3+(Z)/3
z=-2(Y)/3 + Z/3

- subbing this into [1] and multiplying throughout by 3 ,gives any plane maps to:

3a(X) + (Y)(-2a+b-2c) + (Z)(-2a+b+c) [2]

Here I am unsure how to interpret the comparison of the co-efficients between [1] and [2] of x,y and z:

3a=a
b= -2a+b-2c => a =c
c= -2a +b+c => 2a=b

I am unsure of what to do next...
 
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  • #2
any help would be greatly appreciated. The answer is: The plane ax+by+cz=d maps onto itself if a=c and 2a=b.
 

Related to Matrix Transformation - A plane mapping onto itself.

1. What is matrix transformation?

Matrix transformation is a mathematical process where a matrix is used to transform one set of coordinates into another set of coordinates. It is commonly used in computer graphics and image processing.

2. What is a plane mapping?

A plane mapping is a transformation that maps points from one plane onto another plane. It can be thought of as a way of stretching, rotating, or shifting the original plane to a different position or shape.

3. How does a plane mapping onto itself work?

A plane mapping onto itself means that the original plane is being transformed in a way that it remains the same plane after the transformation. This can be achieved by using a special type of matrix called an identity matrix, which does not change the coordinates of the points in the original plane.

4. What is the purpose of a matrix transformation?

The purpose of a matrix transformation is to change the position, shape, or orientation of an object or set of points. It is used in various fields such as computer graphics, image processing, and physics to model and analyze transformations in a more efficient and accurate way.

5. How do you perform a matrix transformation?

To perform a matrix transformation, you need to have a matrix representing the transformation and the coordinates of the points to be transformed. The coordinates are multiplied by the matrix, resulting in the transformed coordinates. The resulting coordinates can then be plotted to visualize the transformation.

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