But what would the idea of scaling an area by a negative amount even mean?

但是将面积按负比例缩放到底是什么意思呢？

The absolute value of the determinant, though still tells you the factor by which areas have been scaled.

行列式的绝对值 仍然告诉你面积被缩放的因子。

The determinant tells you how much a transformation scales areas and eigen vectors are the ones that stay on their own span during a transformation.

行列式告诉你变换对面积的缩放程度特征向量是在变换过程中保持在自己张成的空间上的向量。

Since this region started out with area one and ended up with area six, we can say the linear transformation has scaled its area by a factor of six.

因为这个区域从面积1开始到面积6结束 我们可以说这个线性变换把它的面积扩大了6倍。

Something else you might notice is that if you were to scale up one of those vectors, perhaps multiplying V by three, then the area of that parallelogram is also scaled up by a factor of three.

你可能还会注意到 如果你要将其中一个向量放大 比如将V乘以3 那么平行四边形的面积也会放大3倍。