It turns out that scalars also share this coordinate invariance property.

事实证明，标也具有种坐标不变性。

And what are vectors? Scalars? And what does this mean?

什么是向？标？是什么意思？

The way you'll often hear this described is that linear transformations preserve the operations of vector edition and scalar multiplication.

您经常听到的描述是 线性变换保留了向编辑和标法的操作。

I want you to think about each coordinate as a scaler, meaning think about how each one stretches or squishes.

我想让你们把每个坐标看作一个标 也就是说 想想每个坐标是如何拉伸或挤压的。

Remember how I said that linear algebra revolves around vector edition and scalar multiplication?

还记得我说过线性代数围绕着向编辑和标法吗？

The distance between you and a bench, and the volume and temperature of the beverage in your cup are all described by scalars.

你和长凳之间的距离，以及你杯子里饮料的体积和温度，都是用标来描述的。

Which two dimensional vectors can you reach by altering the choices of scalers?

通过改变标的选择可以得到哪些二维向？

It's a little awkward to work with at 1st, because that left hand side represents matrix factor multiplication, but the right hand side here is scalar vector multiplication.

首先个有点尴尬 因为左边表示矩阵因子的法 但右边是标向的法。

And whenever you catch a number like two or one, 3rd or negative 1.8 acting like this, scaling some vector, you call it a scaler.

当你捕捉到像2或1 3或-1 8样的数 缩放某个向 你称它为标。

Now, the more linear, algebra oriented way to describe coordinates is to think of each of these numbers as a scaler, a thing that stretches or squishes vectors.

现在 更线性的 代数化的描述坐标的方法是把每一个数字看作一个标 一个拉伸或压缩向的东西。

What this means basically is that when you think about coordinates as scalers, the basis factors are what those scalers actually, you know, scale.

基本上意味着 当你把坐标看作标时 基因子就是些标所占的比例。

So in the conception of vectors as lists of numbers, multiplying a given vector by a scaler means multiplying each one of those components by that scaler.

所以在向作为一串数字的概念中 一个给定向以一个标意味着每一个分都以个标。

Our concept of the character is actually a composition of multiple Unicode scalar values, and decomposing these correctly is critical to getting that fidelity with Unicode, the characters, and the string API.

我们对于字符的概念实际上是由多个统一码标值构成的，正确地把它解构对于统一码的精确性以及字符和字符串 API 的精确性都至关重要。

Now, if you let both scalers range freely and consider every possible vector that you can get, there are two things that can happen for most pairs of vectors.

现在 如果你让两个标自由变换考虑你能得到的每一个可能的向 对于大多数向对 有两种情况可能发生。

But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it like a scalar.

但是一些特殊的向仍然在它们自己张成的空间上 意味着矩阵对样一个向的作用就是拉伸它或者像挤压标一样挤压它。

You'll see in the following videos what I mean when I say that linear algebra topics tend to revolve around these two fundamental operations vector edition and scalar multiplication.

在接下来的视频中 你会看到我说线性代数的主题往往围绕着两个基本运算向变换和标法。

Take a moment to think about all the different vectors that you can get by choosing two scalers, using each one to scale one of the vectors, than adding together, what you get.

花点时间想想所有不同的向你可以通过选择两个标 用每个标缩放其中一个向 而不是相加 得到什么。

You'll choose three different scalers, scale each of those vectors, and then add them all together, and again, the span of these vectors is the set of all possible linear combination.

你会选择三个不同的标 缩放每一个向 然后把它们加在一起 同样 些向张成的空间是所有可能的线性组合的集合。

You can kind of imagine turning two different knobs to change the two scalers, defining the linear combination, adding the scaled vectors, and following the tip of the resulting vector.

你可以想象一下 转动两个不同的旋钮来改变两个标 定义线性组合 将缩放的向相加 然后沿着得到的向的尖端移动。

Another way to think about it is that you're making full use of the three freely changing scalers that you have at your own disposal to access the full three dimensions of space.

另一种思考方法是你充分利用了三个自由变换的标你可以随意使用它们来访问空间的全部三维空间。