# Find if a linear transformation is flattening

#### NitrousUK

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
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.

#### NitrousUK

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
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.