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Linear Algebra for Machine Learning and Data Science
34h5m
In week 1 of this course, you'll learn what singular and non-singular matrices are. Since matrices correspond to linear transformations, then linear transformations can also be singular and non-singular. And you'll be pleasantly surprised to see that there is a very nice way to identify linear transformations that are singular and non-singular. Even the rank of a linear transformation can be identified easily. Let me show you how. You already saw previously how the matrix with entries 3, 1, 1, and 2 corresponds to the linear transformation that sends the bases on the left to the bases on the right, and equivalently sends this blue grid to this orange grid over here. This is also called the change of bases since it sends the blue square bases on the left into the orange parallelogram bases on the right. But one special thing is this. The orange grid covers every single point on the plane. It turns out that this is precisely the trait of a non-singular transformation. If the resulting points after multiplying them by a matrix cover the entire plane, then the transformation is non-singular, and vice versa. The points that are covered on the right are called the image of the transformation. So to see this, let's look at an example of a singular matrix such as this one, the matrix with entries 1, 1, 2, 2. Notice that if you multiply the matrix by the vector 0, 0, you get 0, 0, as usual. If you multiply it by the vector 1, 0, you get 1, 2. Now notice what happens if I multiply it by the vector 0, 1. I get again 1, 2, and if I multiply it by 1, 1, I get 2, 4. Which means that the square bases on the left doesn't go to a parallelogram, or it does, but it's a degenerate parallelogram. It's actually a line segment. And there's a problem because the grid on the left doesn't get sent to the entire plane. It gets sent to this line. Because when we had a parallelogram, no matter how thin, it was able to cover the entire plane on the right. But if our parallelogram is flat, and it's only one line segment, then that can only cover the orange line on the right. It cannot cover the entire plane. So, in summary, when a matrix is singular, this thing happens. You're not covering the entire plane on the right. You're only covering a small portion of it. In this case, you're covering a line. Now let's look at a very singular matrix with entries 0, 0, 0, 0. This matrix has any vector with any coordinates into the vector 0, 0, which is the origin. So the entire plane gets sent to that orange point in the origin. So that matrix is definitely very singular because the image is not even a plane. It's not even a line. It's actually a point. So here's a summary. The first matrix sends the whole plane to the whole plane, so it is non-singular. The second one sends the whole plane to simply a line, so it's singular. And the third one sends the whole plane to a point, so it's even more singular. Notice that the dimension of the image of the first one is 2. And that's precisely the rank of this matrix. The dimension of the image here is 1 because it's a line, and that's the rank of this matrix. And the dimension here of the point is 0, which is the rank of the third matrix. So that's another way to calculate rank. The dimension of the image of the linear transformation.