Take the parallelogram defined by the 1st basis factor I had, and the mystery input vector X-Y.

取由第一个基因子定义的平行四边形 以及未知的输入向量X-Y。

Look at the parallelogram we defined earlier, which encodes the excordinate of the mystery input vector spanned by that vector.

看看我们之前定义的平行四边形 它编码了神秘输入向量由该向量张成的横坐标。

Well, we know the coordinates for where the mystery input vector lens.

我们知道神秘输入向量透镜的坐标。

Is there an alternate geometric understanding for the coordinates of our input vector that remains unchanged after the transformation.

是否有另一种几何形式来理解输入向量的坐标在变换之后保持不变。

And what we're really doing is thinking about input vectors moving over to their corresponding outputs.

我们真正在做的是考虑输入向量移动到它们对应的输出。

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。

If a transformation takes some input vector to some output vector, we imagine that input vector moving over to the output vector.

如果一个变换从某个输入向量到某个输出向量 我们可以想象输入向量移动到输出向量。

So what we have is a sort of puzzle which input vector, X-Y is going to land on this output, negative for negative two.

所以我们有一个难题 哪个输入向量X-Y会落在这个输出向量上 对-2是负的。

The area of this parallelogram is the base one times the height perpendicular to that base, which is the Y coordinate of that input vector.

这个平行四边形的面积等于底1乘以垂直于底的高 也就是输入向量的Y坐标。

Well, this is the transformed version of the parallelogram we were looking at earlier, the one whose area was the Y coordinate of the mystery input vector.

这是我们之前看到的平行四边形的变换后的版本 它的面积是未知输入向量的Y坐标。

Then to understand the transformation as a whole, we might imagine watching every possible input vector move over to its corresponding output vector.

然后 为了从整体上理解这个变换 我们可以想象看到每一个可能的输入向量移动到它对应的输出向量。

And symmetrically, if you look at the parallelogram, spend by our input vector, and the 2nd basis, j hat, its area is going to be the ex coordinate of that mysteryfector.

对称地 如果你观察这个平行四边形 花在我们的输入向量上 第二个基底 j帽 它的面积就是这个神秘因子的前坐标。

That way to think about a transformation, taking every possible input vector to some output vector, we watch every point in space moving to some other point, in the case of transformations in two dimensions.

用这种方式来思考一个变换 把每一个可能的输入向量变成某个输出向量 我们观察空间中的每一个点移动到另一个点 在二维变换的情况下。

And if you think of, say, the Z coordinate of that input vector as the volume of that special parallela pipe ed we were looking at earlier, spanned by I hat j Hat, and the mystery input vector.

如果你把输入向量的Z坐标想象成我们之前看到的那个特殊的平行管道的体积 由I*j*张成 还有那个神秘的输入向量。

Geometrically, the meaning of this function is that for any input vector X-Y-Z, you consider the parallelopyped defined by this vector, V and W, then you return its volume with a plus or minus sign, depending on orientation.

几何上 这个函数的意义是 对于任何输入向量X-Y-Z 你考虑由这个向量V和W定义的平行体 然后你返回它的体积 根据方向带正号或负号。