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。

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。

Each of those coordinates can be thought of as instructions for how to scale each basis vector so that they add together to get your vector.

每一个坐标都可以看作是如何缩放每个基向量的指令这样它们相加就得到了向量。

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。

What this means is that the particular way to get to that vector using her two basis vectors is to scale B one by five thirds, scale B two by one, 3rd, then add them both together.

这意味着用两个基向量得到这个向量的特殊方法是将B缩放为1×5/3 将B缩放为2×1/3 然后将它们相加。