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。

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。

So in that very special case, X would be the dot product of the 1st column with the output vector, and why would be the dot product of the 2nd column with the output vector?

在这种特殊情况下 X是第一列与输出向量的点积 为什么是第二列与输出向量的点积呢？

So what you might do is create a new matrix whose 1st column is the same as that of our matrix, but whose 2nd column is the output vector, and then you take its determinant.

那么你要做的就是创建一个新矩阵它的第一列和我们的矩阵一样 但是它的第二列是输出向量 然后你求它的行列式。

And j Hat, the transformed version of this guy, is spanned by the output vector, and the 2nd column of the matrix, and its area will have been multiplied by the determinant of that matrix.

而j帽 这个向量的变换后的形式 是由输出向量张成的 和矩阵的第二列 它的面积乘以这个矩阵的行列式。

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。