And then all of the basis vectors, other than the one, correspond to the direction you're looking for.

然后所有的基向量 除了这个 都对应于你要找的方向。

For this specific example, the basis vector I hat is one such special vector.

对于这个特殊的例子 基向量I帽就是这样一个特殊的向量。

A set of basis vectors, which are also agen vectors, is called, again, reasonably enough, an eigen basis.

一组基向量 也就是原向量 再一次 很合理地称为特征基。

Those columns represent were our basis, vectors.

这些列表示我们的基底 向量。

Take a look at what happens if our basis vectors just so happened to be eigenvectors, e.g.

看一下如果基向量恰好是特征向量会发生什么。

What I'd like to talk about here is the idea of using a different set of basis vectors.

我在这里要讲的是使用一组不同的基向量。

And the two special vectors I had and j hat are called the basis vectors of our standard coordinate system.

这两个特殊的向量和j帽叫做标准坐标系的基向量。

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它和第一个基向量(0)的点积。

A matrix whose columns represent jennifer's basis factors can be thought of as a transformation that moves our basis vectors.

一个列代表jennifer基因子的矩阵可以被认为是移动基向量的变换。

We could have chosen different basis vectors and gotten a completely reasonable new coordinate system.

我们可以选择不同的基向量得到一个完全合理的新坐标系。

And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigen values.

解释这个的方法是所有的基向量都是特征向量 这个矩阵的对角线元素是它们的特征值。

But now there are three standard basis vectors that we typically use.

但是现在有三个标准基向量我们通常使用。

Delightfully, these transformations can be described using only a handful of numbers, the coordinates of where each basis vector lands.

令人高兴的是 这些转换可以只用几个数字来描述 即每个基向量的落点坐标。

This is because it represents working in a coordinate system, where what happens to the basis vectors is that they get scaled during the transformation.

这是因为它表示在一个坐标系中工作 基向量在变换过程中会缩放。

It means the basis vectors continue to span the full two dimensions of space, and the determinant is not zero.

它意味着基向量继续张成整个二维空间 行列式不为零。

And just as with two dimensions, one of these transformations is completely described by where the basis vectors go.

就像在二维空间中一样 其中一个变换完全由基向量的位置来描述。

Now, take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates three, two, using our basis vectors I had.

现在 再看一下我之前展示过的向量 你和我将用坐标来描述的向量 用我们已知的基向量。

So maybe you hope that after the transformation, the dot products with the transformed version of the mystery vector, with the transformed version of the basis vectors, will also be these coordinates X and y.

所以也许你希望在变换之后 与神秘向量的变换后的点积 与基向量的变换后的点积 也会是这些坐标X和y。

In three dimensions, it helps to focus your attention on the specific one by one by one cube whose edges are resting on the basis vectors.

在三维空间中 它有助于你把注意力集中在一个接一个的特定立方体上 这些立方体的边都位于基向量上。

One of the most important consequences of these properties, which makes matrix factor multiplication possible, is that a linear transformation is completely described by where it takes the basis vectors.

这些性质的最重要的结果之一 使得矩阵因子乘法成为可能 就是线性变换完全由它取基向量的位置来描述。