But what we want is to figure out what exactly that linear combination should be.

但我们想要的是找出这个线性组合到底是什么。

A linear combination of three vectors is to find pretty much the same way as it is for two.

三个向量的线性组合和两个向量的方法是一样的。

So any time that you're scaling two vectors and adding them like this, it's called a linear combination of those two vectors.

所以任何时候当你缩放两个向量并像这样相加时 它被称为这两个向量的线性组合。

In other words, adding a scaled version of that 3rd vector to the linear combination doesn't really give you access to any new vectors.

换句话说 在线性组合中加入第三个向量的缩放版本并不能让你得到任何新的向量。

Before the linear transformation, were thinking of this vector as a certain linear combination of our basis factors, negative one times I had plus two times j Hat.

在做线性变换之前 我们认为这个向量是基因子的一个线性组合 1乘以加上2乘以j。

In other words, it started off as a certain linear combination of I had A-J hat, and it ends up as that same linear combination of where those two vectors landed.

换句话说 它开始是a-j帽的某个线性组合 它最终是这两个向量的相同的线性组合。

But isn't it more fun to think about these columns as the transformed versions of your basis vectors, and to think about the results as the appropriate linear combination of those vectors?

但是把这些列看作基向量的变换后的形式 并把结果看作这些向量的适当线性组合不是更有趣吗？

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量 定义线性组合 将缩放的向量相加 然后沿着得到的向量的尖端移动。

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量 缩放每一个向量 然后把它们加在一起 同样 这些向量张成的空间是所有可能的线性组合的集合。

Here's some more terminology. The set of all possible vectors that you can reach with a linear combination of a given pair of vectors is called the span of those two vectors.

这里还有一些术语 所有可能的向量的集合你可以通过一个给定向量对的线性组合得到这个集合叫做这两个向量张成的空间。

Another way of frasing that would be to say that one of the vectors can be expressed as a linear combination of the others, since it's already in the span of the others.

另一种折叠的方法是其中一个向量可以表示成其他向量的线性组合 因为它已经在其他向量张成的空间中。

Well, their span is the collection of all possible linear combinations of those two vectors, meaning all possible vectors you get by scaling each of the two of them in some way and then adding them together.

它们张成的空间是这两个向量所有可能的线性组合的集合 也就是说所有可能的向量都是通过将它们分别缩放然后相加得到的。