It's not at all true for most linear transformations.

这对大多数线性变换都不成立。

If any of that feels uneasy, consider some linear transformation, like a 90 degree counter clockwise rotation.

如果你觉得不舒服 可以考虑一些线性变换 比如逆时针旋转90度。

With this projection, we just defined a linear transformation from two D vectors to numbers.

通过这个投影 我们定义了一个从两个D向量到数字的线性变换。

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 线性变换保留了向量编辑和标量乘法的操作。

There's some kind of connection between linear transformations that take vectors to numbers and vectors themselves.

把向量转化为数的线性变换和向量本身之间有某种联系。

So that performing the linear transformation is the same as taking a dot product with that vector the cross product.

所以进行线性变换就等于对这个向量做点积也就是叉乘。

To start, let's just parse this term linear transformation.

首先 我们来分析一下线性变换这个术语。

Some linear transformations are simple to think about, like rotations about the origin.

有些线性变换很简单 比如绕原点旋转。

In general, you should think of linear transformations as keeping grid lines parallel and evenly spaced.

一般来说 您应该将线性变换视为保持网格线平行且间距均匀。

As with the cases we've seen before, one of these linear transformations is completely determined by where it takes eye hat and j hat.

和我们之前见过的情况一样 其中一个线性变换是完全由它的位置决定的。

E.G. there's a perfectly reasonable notion of a linear transformation for functions, something that takes in one function and turns it into another one.

例如 关于函数的线性变换有一个非常合理的概念 就是把一个函数变成另一个函数。

Since linear transformations must keep the origin fixed in place.

因为线性变换必须保持原点固定。

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。

With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis factors.

对于任何由矩阵描述的线性变换 你可以通过读出矩阵的列作为基因子的落点来理解它在做什么。

Mathematicians reply that it is Dingle who got his sums wrong, not Einstein: that Dingle used linear transformation to derive two different values for the same quantity, which is impossible.

数学家回答说，是丁格尔算错了，而不是爱因斯坦：丁格尔使用线性变换对同一数量得出两个不同的值，这是不可能的。

But linear transformations are, of course, much more restricted than your run of the mail function with a two D input and a one D output.

但是线性变换 当然 比你的邮件函数有更多的限制有二维的输入和一维的输出。

The overall effect here, from start to finish, is another linear transformation distinct from the rotation and the sheer.

这里的整体效果 从开始到结束 是另一种不同于旋转和纯粹的线性转换。

But luckily, linear algebra limits itself to a special type of transformation, ones that are easier to understand, called linear transformations.

但幸运的是 线性代数将自己局限于一种特殊的变换 一种更容易理解的变换 叫做线性变换。

But a fuller understanding of the role that dot products play in math can only really be found under the light of linear transformation.

但要更全面地理解点积在数学中的作用 只有在线性变换的指导下才能真正找到。

What's more, because of the way linear transformations work, any other vector on the x axis is also just stretched by a factor of three, and hence remains on its own span.

更重要的是 由于线性变换的工作方式 x轴上的任何其他向量也只是被拉伸了3倍 因此保持在它自己的张成空间上。