Stretching, Rotating, and Shearing

An interactive visual lab for Chapter 6. Move the sliders, watch the grid, track the basis vectors, and learn to read a matrix as a geometric machine.

1. The matrix machine

A 2 by 2 matrix transforms every point in the plane. The columns tell where the standard basis vectors go.

If A = [[a,b],[c,d]], then the first column is Ae1 and the second column is Ae2.


    

    
  






  
2. Transform the unit square and circle

  

    

      
The unit square becomes the parallelogram spanned by the columns of the matrix. The unit circle shows how the transformation stretches different directions.

      
A rotation keeps the circle circular. A stretch or shear turns it into an ellipse. A collapse turns it into a line segment.

      
Student task. Choose a matrix with determinant close to zero. What happens to the square? What happens to the circle?

    

    
  






  
3. Build transformations by sliders

  

    

      
This panel builds a matrix from a rotation, a horizontal shear, and separate horizontal/vertical scaling:

      
A = Rotation × Shear × Scale

      

        

        

        

        

      

      

    

    
  






  
4. Data cloud transformation

  

    

      
A dataset with two features is a cloud of points. Applying a matrix reshapes the cloud. This is a preview of linear algebra in data science.

      
The same matrix from Panel 1 is applied to the data cloud.

      
Try this. Use the Collapse preset above. Notice how a two-dimensional cloud becomes almost one-dimensional.

    

    
  






  
5. Transform a simple picture

  

    

      
A picture can be represented by points and line segments. Matrix transformations are the language behind many operations in computer graphics.

      
The same matrix from Panel 1 transforms this small house drawing.

      
Question: which transformations preserve the house's shape? Which ones distort it?

    

    
  






  
Reflection prompts

  
    
    
  1. Why do the columns of a matrix determine the whole transformation?
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  2. What geometric information does the determinant give?
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  3. How can a matrix preserve area but still distort angles?
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  4. Why is matrix multiplication the correct operation for doing transformations in sequence?
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