# Pre- and Co-Image (Session 2022)

The format today will differ somewhat from the norm. Instead of having one debrief the very end, we will try to do two exercises with a shorted debrief after each one.

Feel free to look at the debrief notes as hints to solving the exercises; just take a couple of minutes first to try to make sense of the question and make a sketch.

# First Exercise (3.9)

Exercise 3.9 are from Ma 2004 page 62ff.

## Debrief Notes

### Part 1

You should first find the pre-image of the image of $$L$$.

• What kind of object is the pre-image?
• How did we describe such an object previously?
• What is the relationship between this pre-image and a point $$x\in L$$?
• What is the relationship between the pre-image and and the vector $$\ell$$?

### Part 2

• If you read the points $$x^1$$ and $$x^2$$ as vectors in 3D, what do they look like?
• Can you describe the pre-image in terms of $$x^1$$ and $$x^2$$?
• maybe as a span?
• What then is the relationship between $$\ell$$ and $$x^1,x^2\in L$$?

How do you find a vector which is orthogonal on two known vectors in 3D?

### Part 3

• Note that $$x$$ is an image point.
• $$\ell^1$$ and $$\ell^2$$ are vectors in 3D, and co-images of two image lines
• If you view $$x$$ as a 3D vector instead of a point, what does it look like?
• What would be the relationship between this vector $$x$$ and $$\ell^1$$ and $$\ell^2$$?
• How do we find vector $$x$$ with the right relationship with $$\ell^1$$ and $$\ell^2$$?
• How do we make sure that the vector $$x$$ is an image point $$x$$?

# Second Exercise (3.10)

Exercise 3.10 are from Ma 2004 page 62ff.

## Debrief Notes

1. Here, it is necessary to look at the pre-images of the two lines.
• What does the pre-images look like?
• What is the intersection of the pre-images? Could it be empty?
• What is the intersection between the image plane and the pre-images?
2. Here, you need to look at the co-images.
• What can you say about co-images of parallel lines?
• What can you say about the relationship between the co-images and the images? Is there are relationship between one line and the co-image of the other line?