Effects of Linear Transformations on Area#
- Whenever a linear transformation happens, it either stretches the space out or squishes it. See last chapter.
- Now, it can be helpful to note, by how much it stretches or squishes the area.
- Example 1:
For matrix:
$$
\begin{bmatrix}
3 & 0 \\
0 & 2
\end{bmatrix}
$$
If we see how much it scales the unit matrix:
It started out with area 1 and ended up with area 6. So, we can say this linear transformation
has scaled its area by a factor of 6. - Example 2: Shear
You can see in this case that the area remains the same. - So, if we know how much area of that single unit square changes, we can tell how much the area of any possible region in space changes. Because whatever happens to one square in the grid has to happen to any other square in the grid, no matter the size.
- And, any shape that is not a grid square can be approximated using grid squares, if we use
small enough grid squares.
Determinant#
- This scaling factor, the factor by which the linear transformation changes any area is called the Determinant of that transformation.
- Example 1:
Here you can see the determinant of the transformation is 3 because it increases the
area by a factor of 3. - Example 2:
Here you can see the determinant of the transformation is 0.5 because it decreases the
area by a factor of 0.5. - Example 3:
Here you can see the determinant of the transformation is 0 because it squishes all of
space into a line or a single point(in both cases the area will become 0).NOTE: This example is quite important. It says that, checking to see if determinant of a given matrix is 0, will give a way of computing whether or not transformation associated with that matrix, squishes everything into a smaller dimension.
- Negative determinant:
The negative determinant has to do with the orientation of the space. So, if after a transformation, the orientation of space have been inverted, that would make the determinant negative.
By orientation has been reversed what we mean is that let’s think about in terms of \( \hat{\imath} \) and \( \hat{\jmath} \).
Here, you can see initially \( \hat{\jmath} \) was left of \( \hat{\imath} \) and after the transformation it is to the right of \( \hat{\imath} \). So, this would make the determinant negative, because the orientation of space has been reversed.
Another example:
Visualizing determinants:
Determinants in 3D#
- Instead of area, this time we see how much does the volume of the 3D unit cube scales:
This shape is called a Parallelepiped.
Since, the volume of the unit cube was 1, we can think of the determinant as the volume
of the parallelepiped, that the cube turns into. - So, if a determinant in 3D is zero we can say the columns are linearly dependent.
- Negative Determinants in 3D:
To do this we need to use the right hand rule.
If after a transformation you can still use your right hand, the orientation has not changed,
and the determinant is positive. But, if you have to use your left hand, the orientation has changed,
and the determinant is negative.
Formula to calculate determinants#
- For 2x2 matrices: $$ \text{det}\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc $$
- Derivation of the formula:
$$
\text{det}\begin{bmatrix}
a & b \\
c & d
\end{bmatrix} = (a + b)(c + d) - ac - bd - 2bc = ad - bc
$$ - For 3x3 matrices: $$ \text{det}\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} = a \ \text{det}\begin{bmatrix} e & f \\ h & i \end{bmatrix} - b \ \text{det}\begin{bmatrix} d & f \\ g & i \end{bmatrix} + c \ \text{det}\begin{bmatrix} d & e \\ g & h \end{bmatrix} $$
Why does \( \det(M_1M_2) = \det(M_1) \det(M_2) \)?#
The determinant of a matrix measures how much space is scaled (area in 2D, volume in 3D) by the linear transformation that the matrix represents.
So when you multiply two matrices \( M_1M_2 \), you’re applying \( M_2 \) first (scaling space by \(\det(M_2)\)), then \( M_1 \) (scaling it again by \(\det(M_1)\)).
Since, scaling effects multiply — just like scaling a shape by 2 and then by 3 scales it by \(2×3=6\) — the total scaling is:
$$ \det(M_1M_2) = \det(M_1) \det(M_2) $$