We may notice that there are some 「Linear Transformations」 seem to 「stretch space out」 while others 「squish it space in」 我们可能注意到有些「线性变换」会「将空间向外拉伸」，有的则「将空间向内挤压」
it is important to 「measure exactly how much」 a given transformation 「stretches and squishes space」. 「测量」变换究竟对空间「有多少拉伸或挤压」是一件很重要的事情
More specifically, to 「measure the factor」 by which 「the area of a given region increases or decreases」. 更具体一点，就是「测量一个给定区域面积增大或减小的比例」
Example
For example look at the matrix with the columns and
比如说这样一个以和为列的矩阵
It scales by a factor of and scales by a factor of 它将 伸长为原来的 倍，将 伸长为原来的 倍
Since this region 「started out with area 1」, and 「ended up with area 2 x 3 = 6」 因为这个区域「初始面积为1」，「最终面积为2 x 3 = 6」
the 「linear transformation」 has 「scaled it's area by a factor of 6」. 所以我们说这个「线性变换」将它的「面积变为6倍」
2. Understand Determinant
The 「scaling factor」, by which a linear transformation 「changes any area」 is called the 「determinant of that transformation」. 这个「缩放比例」，即「线性变换改变面积的比例」,被称为「这个变换的行列式」
the 「determinant」 of a transformation would be 「3」 if that transformation 「increases the area of the region by a factor of 3」. 一个线性变换的「行列式是3」就是说它「将一个区域的面积增加为原来的3倍」
the 「determinant」 of a transformation would be 「1/2」 if it 「squishes down all areas by a factor of 1/2」. 一个线性变换的「行列式是1/2」就是说它「将一个区域的面积缩小一半」
The Meaning of Determinant = 0
If 「the determinant」 of a 2D transformation 「equals to 0」, this means that 「it squishes all of space onto a line or even onto a single point」 如果一个二维线性变换的「行列式为0」，说明「它将整个平面压缩到一条线，甚至是一个点上」
since 「the area of any region would become 0」. 因为此时「任何区域的面积都变成了0」
checking 「if the determinant of a given matrix is 0」 will check 「whether the transformation associated with that matrix squishes」 everything into a 「smaller dimension」. 只需要检验一个「矩阵的行列式是否为 0」 我们就能了解「这个矩阵所代表的变换是否」将空间「压缩」到「更小的维度上」
The Meaning of Determinant < 0
「Scaling an area by a Negative
Determinant」 means to 「turn over that 2D plane」 onto the 「other side」. 将一个区域「缩放负数倍」意味着「将二维平面翻转」到了「另一面」
the 「orientation of space」 has been 「inverted」. 「空间定向」就发生了「翻转」
the 「absolute value」 of the determinant 「still tells you the factor」 by which 「areas have been scaled」. 行列式的「绝对值」依然表示「区域面积的缩放比例」
Application
Consider a scenario where 「a robotic arm is tasked to position a cup of water at a specific coordinate in 3D space」. 设想一个情景，「机器人手臂在三维空间中将一杯水定位到特定的坐标」。
During this process, we not only expect the robot to 「accurately execute the operation through 3D spatial transformations」, but it's also crucial that 「the sign of the determinant remains consistent」. 在这个过程中，我们不仅期望机器人能「通过三维空间变换精确执行操作」，更关键的是「行列式的符号需要保持一致」。
A 「change in the sign of the determinant」 would 「indicate a flip in the orientation of space」, leading to the 「overturning of the water cup」. 如果「行列式的符号出现改变」，那将意味着「空间的定向发生了翻转」，这将导致「水杯翻倒」。
Thus, the 「sign of the determinant is essentially a key factor in maintaining spatial orientation」, ensuring the 「stability」 of the operational object, such as a cup of water, during spatial transformations. 因此，「行列式的符号实质上是维护空间定向的重要因素」，保证了操作对象，如水杯，在空间变换中的「稳定性」。
3. Compute Determinant
2 x 2 Matrix
The formula for a 2 by 2 matrix:
if and 「both are 0」, then tells how much is stretched in the and tells how much is stretched in the .
it should make sense that gives 「the area of the rectangle」 that 「the unit square turns into」.
if both and are 「NOT equal to 0」, then tells 「how much this parallelogram」 is 「stretched or squished」 in the 「diagonal direction」.
which shows how much is changed in the and how much is changed in the
3 x 3 Matrix
The formula for a 3 by 3 matrix:
A Property
If we 「multiply 2 matrices together」, the 「determinant of the resulting matrix」 is the same as the 「product of two determinants of the original two matrices」
since we first apple which 「scales space」 by 「」 and then apply which 「scales space」 by 「」
this is same as we apply both of them which 「scales space」 by 「」
