Vectors & Matrices Defined Up to Scale

In summary, "defined up to scale" means that in homogenous coordinates, two vectors that are multiples of each other are considered to be the same vector. This is important in the context of geometric camera models and the equation of a plane, where the vector representing the plane is only defined up to scale. This terminology is not commonly used in elementary linear algebra textbooks.
  • #1
gnome
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What does it mean for a matrix or a vector to be "defined up to scale"?

I don't remember ever seeing this expression in my very limited exposure to linear algebra. To put it in context, I'm finding it in a text on computer vision in the section on geometric camera models.
They're talking about using something called homogeneous coordinates to represent points, vectors and planes, and they put the equation of a plane:
[tex]ax + by + cz -d = 0[/tex]
as the dot product of two vectors:
[tex]\Pi \cdot P = 0[/tex] (Eq. 2.2)
where
[tex]\Pi = \left ( \begin{array}{c} a\\ b\\ c\\ -d \end{array} \right ) \; \text{and} \; P = \left ( \begin{array}{c} x\\ y\\ z\\ 1 \end{array} \right ) [/tex]

Anyway, it goes on to say
Note that [itex]\Pi[/itex] is only defined up to scale since multiplying this vector by any nonzero constant does not change the solutions of Eq. 2.2. We use the convention that homogeneous coordinates are only defined up to scale, whether they represent points or planes...
I don't understand what is accomplished by putting the equation in this form, nor do I understand the significance of "defined up to scale". I don't see this terminology anyplace in my (elementary) linear algebra textbook (or I don't know what to look for). Any idea where I can find a clear explanation?


(Why did I write "matrix question" as the title of this thread? I only mentioned vectors, not matrices, in my question, but this section of the text also deals with many matrices that are "only defined up to scale".)
 
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  • #2
In "homogenous coordinates", two "vectors", where one is a multiple of the other, are considered to be the same vector.

That is, <1, 2, 3, 1> , <2, 4, 6, 2>, and <3, 6, 9, 3> are all different representations of the same (3 dimensional) vector. In terms of ordinary 4 dimensional coordinates, they would, of course, represent vectors having the same direction but different lengths- hence, the three dimensional vector they all represent is "defined up to scale".
 
  • #3
Thanks HoI. I was looking in all the wrong places. I see there's quite a bit of info on "homogenous coordinates" on the web.
 

1. What is a vector?

A vector is a mathematical object that represents both magnitude and direction. It can be thought of as an arrow in space, with a specified length and direction.

2. What are the components of a vector?

A vector has two main components: magnitude and direction. The magnitude is the length of the vector, while the direction is the angle at which the vector is pointing.

3. What is a matrix?

A matrix is a rectangular array of numbers or symbols that are arranged in rows and columns. It is often used to represent data or to perform mathematical operations.

4. How are vectors and matrices related?

Vectors and matrices are closely related as vectors can be thought of as a special case of a matrix with only one row or column. Matrices can also be used to perform operations on vectors, such as scaling or rotating.

5. What does it mean for a vector or matrix to be defined up to scale?

When a vector or matrix is defined up to scale, it means that the values of the vector or matrix can be multiplied by a constant without changing the overall meaning or properties of the vector or matrix. This is often used in linear algebra to simplify calculations and to focus on the underlying structure of the data.

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