Solving a linear system with an orthonormal matrix is actually super easy, because dot products are preserved.

用标准正交矩阵求解线性方程组其实非常简单 因为点积被保留了。

That is, they don't preserve that zero dot product.

也就是 不保留0点积。

The dot product before and after the transformation will look very different.

变换前后的点积看起来很不一样。

And looking at the example I just showed, dot products certainly aren't preserved.

看看我刚才举的例子 点积肯定不会被保留。

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

相关的背景知识是理解行列式 一点点积 当还有线性方程组。

In the sense that applying the transformation is the same thing as taking a dot product with that vector.

在这个意义上应用这个变换就相当于对这个向量做点积。

In fact, worthwhile side note here transformations which do preserve dot products are special enough to have their own name.

事实上 值得注意的是保留点积的变换很特别 有自己的名。

And if they point in generally the opposite direction, their dot product is negative.

通常指向相反的方向 的点积是负的。

The excordinate of this mystery input vector is what you get by taking its dot product with the 1st basis vector one zero.

这个神秘的输入向量的纵坐标就是和第一个基向量(0)的点积。

Likewise, things that start off perpendicular with dot product zero, like the two basis factors, quite often don't stay perpendicular to each other after the transformation.

同样地 开始与点积0垂直的东西 比两个基因子 在变换之后通常不会相互垂直。

When their perpendicular meaning, the projection of one onto the other, is the zero vector, their dot product is zero.

当垂直的意思 一个向量在另一个向量上的投影 是零向量时 的点积是零。

And that's probably the most important thing for you to remember about the dot product.

这可能是你要记住的关于点积最重要的一点。

Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.

在这之后 我会给你一些关于点积的知识 还有一些很酷的东西当你在线性变换的光照下看的时候。

So that wraps up dot products and cross products.

这就包含了点积和叉乘。

So that performing the linear transformation is the same as taking a dot product with that vector the cross product.

所以进行线性变换就等于对这个向量做点积也就是叉乘。

Now, this numerical operation of multiplying a one by two matrix by a vector feels just like taking the dot product of two vectors.

现在 这个1×2矩阵乘以一个向量的数值运算就像取两个向量的点积。

So when two vectors are generally pointing in the same direction, their dot product is positive.

所以当两个向量通常指向同一个方向时 的点积是正的。

Ordinarily, the way you might think about one of a vector's coordinates tape its Z coordinate, would be to take its dot product with the 3rd standard basis factor, often called k hat.

通常 你可能会认为一个向量的坐标带的Z坐标 就是取与第三个标准基因子的点积 通常称为k帽。

So in that very special case, X would be the dot product of the 1st column with the output vector, and why would be the dot product of the 2nd column with the output vector?

在这种特殊情况下 X是第一列与输出向量的点积 为什么是第二列与输出向量的点积呢？

Learn what you have learned, and imagine that you don't already know that the dot product relates to projection.

学习你学过的知识 想象一下你还不知道点积和投影的关系。