Find if a linear transformation is flattening

Feb 2009
11
0
I don't know if there's another term for the type of linear transform that collapses all points onto a line or the origin, but my book calls it a "flattening", and I'm wondering how you can tell if a transform is of a flattening type?
The book I have says if one column of the matrix for the transform is a "scalar multiple" of the other, then it's flattening. Not quite sure what that means, however the example uses the matrix:

2 3
4 6

And says the columns have a difference of 2/3 which means it's flattening, which confuses me as I thought scalar meant a whole integer? Any help/hints in this would be massively appreciated!!
 

Opalg

MHF Hall of Honor
Aug 2007
4,039
2,789
Leeds, UK
I don't know if there's another term for the type of linear transform that collapses all points onto a line or the origin, but my book calls it a "flattening", and I'm wondering how you can tell if a transform is of a flattening type?
The book I have says if one column of the matrix for the transform is a "scalar multiple" of the other, then it's flattening. Not quite sure what that means, however the example uses the matrix:

2 3
4 6

And says the columns have a difference of 2/3 which means it's flattening, which confuses me as I thought scalar meant a whole integer? Any help/hints in this would be massively appreciated!!
A scalar is just a number (as opposed to, say, a vector or a matrix). It doesn't have to be a whole number. So 2/3, \(\displaystyle \sqrt2\) and \(\displaystyle \pi\) are all scalars.

The term "flattening" is not standard, but if the book chooses to uses, that's fine. (A mathematician would probably use the term singular matrix.) For a 2×2 matrix, there are various equivalent ways of defining this property. If you have met determinants, the simplest definition is probably to say that a matrix is flattening if its determinant is 0. Otherwise, you can use the definition in the book, that one column should be a scalar multiple of the other. In the case of the matrix \(\displaystyle \begin{bmatrix}2&3\\4&6\end{bmatrix}\), each element in the left column is 2/3rds of the corresponding element in the right column. Another equivalent definition is that one row should be a scalar multiple of the other. In the case of that matrix, the bottom row is twice the top row.
 
Feb 2009
11
0
Ah! Thanks!
It really doesn't help my course chooses to use a lot of non-standard terminology, and not explain particularly well (Open University). I see now it means each element in one column being the same multiple of the corresponding element in the other column.
Thanks for clearing that up!
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
The determinant of the matrix \(\displaystyle \begin{bmatrix}a & b \\ c & d\end{bmatrix}\) is ad- bc. If that equal 0, then ad- bc= 0 which is the same as ad= bc and, if b and d are not 0, \(\displaystyle \frac{a}{b}= \frac{c}{d}\) which says that one column is a multiple of the other.