That is, if it has a zero determinant.

说 如果它的行列式为零。

So its area is just going to be the determinant of the transformation, multiplied by that, why coordinate?

所以它的面积这个换的行列式 乘以这个 为什么要坐标？

The relevant background here is understanding determinance, little bit of dot products, and of course, linear systems of equations.

相关的识理解行列式 一点点积 当然还有线性方程组。

Zero. Subtract off lambder from the diagonal elements and look for when the determinant is zero.

零 从对角线元素中减去lambder然后寻找行列式为零的时候。

As that value of lamb to changes, the matrix itself changes, and so the determinant of the matrix changes.

当lamb的值改时 矩阵本身会改 所以矩阵的行列式会改。

How specifically you think about computing that determinant is kind of beside the point.

如何计算行列式不重点。

Or rather, we should be talking about the signed volume of paralleloppipets in the sense described in the determinant video, using the right hand rule.

或者说 我们应该讨论平行体的带符号体积在行列式视频中描述过 用右手定则。

But, and this is the key idea of determinance, all of the areas get scaled by the same amount, namely, the determinant of our transformation matrix.

但 这行列式的关键思想 所有的面积都被等量缩放 换矩阵的行列式。

Vectors, core topics in linear algebra, like determinants and eigenvectors, seem indifferent to your choice of coordinate systems.

向量 线性代数的核心主题 像行列式和特征向量 似乎与你选择的坐标系无关。

The determinant is all about measuring how areas change due to a transformation.

行列式关于测量面积如何因换而化的。

Then you compute the determinant of this matrix.

然后计算这个矩阵的行列式。

There's no nonsense with basis factors stuck in a matrix or anything like that, just an ordinary determinant returning a number geometrically.

在矩阵中没有基因子之类的废话 只一个普通的行列式几何上返回一个数。

It's not just a coincidence that the determinant is once again important.

行列式很重要 这不巧合。

And actually, to be a little more, you should think of this as the signed area of that parallelogram in the since described in the determinant video.

实际上 更确切地说 你应该把这个看做平行四边形的带符号面积在行列式视频中描述过。

Wrapping your mind around this concept is going to help consolidate ideas from linear algebra, like the determinant and linear systems, by seeing how they relate to each other.

把你的思维围绕在这个概念上有助于巩固线性代数的概念 比如行列式和线性系统 通过观察它们之间的关系。

I'll actually leave it to you to work through the details of why this is true based on properties of the determinant.

我会把它留给你们去研究为什么这个正确的基于行列式的性质。

So that means we can solve for why by taking the area of this new parallelogram in the output space divided by the determinant of the full transformation.

这意味着我们可以通过用输出空间中这个新的平行四边形的面积除以完整换的行列式来解决问题。

The goal here is to find a value of Lambda that will make this determinant zero, meaning the tweaked transformation squishes space into a lower dimension.

这里的目标找到使这个行列式为零的Lambda值 这意味着调整后的转换将空间压缩到更低的维度。

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.

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

So for this video, I'm assuming that everybody has watched Chapter five on the Determinant, and chapter seven, where I introduced the idea of duality.

在这个视频中 我假设每个人都看过第五章的行列式 第七章 我介绍了对偶的概念。