The most important kind of magical step is laying them out in a matrix and doing what's called matrix multiplication.

最关键的神奇步骤是将它们排列成矩阵，并进行所谓的矩阵乘法。

Applying One transformation after another is captured algebraically with matrix multiplication.

应用一个接一个的变换是用矩阵乘法代数捕获的。

Once you know that it's linear, you know that there's some way to describe this function as matrix multiplication.

一旦你知道它是线性的 你就知道有一些方法可以用矩阵乘法来描述这个函数。

But when you think about matrix multiplication as applying one transformation after another, this property is just trivial.

但是当你把矩阵乘法看作是应用一个又一个变换时 这个性质就很简单了。

Easier to think about rotations performing This matrix multiplication numerically is, once again, pretty similar to the two dimensional case.

更容易考虑旋转执行这个矩阵乘法的数值运算 再一次 和二维情况很相似。

Trust me, this will give you a much better conceptual framework that makes the properties of matrix multiplication much easier to understand.

相信我 这将给你一个更好的概念框架 使矩阵乘法的性质更容易理解。

For this next part, it's important that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication corresponds to composing success of transformations.

在接下来的部分中 重要的是你们都能自如地用矩阵表示变换 并且知道矩阵乘法是如何与成功组合变换相对应的。

No matrix multiplication necessary. I remember when I 1st took linear algebra, there was this one homework problem that asks us to prove that matrix multiplication is associated.

不需要矩阵乘法 我记得我第一次学线性代数的时候 有一道作业要求我们证明矩阵乘法是相关的。

This is an honest goodness proof that matrix multiplication is associated, and even better than that, it's a good explanation for why that property should be true.

这是一个很好的证明矩阵乘法是相关的 甚至比这更好 它很好地解释了为什么这个性质是正确的。

But I really do think that before memorizing that process, you should get in the habit of thinking about what matrix multiplication really represents, applying one transformation after another.

但我真的认为在记住这个过程之前 你应该养成思考矩阵乘法真正代表什么的习惯 应用一个又一个变换。