But that vector you had is a two dimensional vector living in the input space.

但是这个向量是一个二维的向量 存在于输入空间中。

But the matrix is still full rank, since the number of dimensions in this column space is the same as the number of dimensions of the input space.

但是这个矩阵仍然是满秩的 因为这个列空间的维数和输入空间的维数是一样的。

What I have pictured here is the input space on the left, which is just two D space, and the output of the transformation shown on the right.

我在这里画的是左边的输入空间 它只是二维空间 右边是变换的输出。

Remember, the columns of this matrix are telling you how that matrix acts as a transform, each one telling you where the basis factors of the input space land.

记住 这个矩阵的列告诉你这个矩阵是如何进行变换的 每一列都告诉你输入空间的基因子在哪里。