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Given a particular matrix, if you want to compute its trace, you will be basically adding up the diagonal entries. But this is quite an algebraic approach in thinking about trace. Can we visualise whats really happening when we add these diagonal entries? The visualisation will also explain many properties of trace you might have seen. There is one particular property that Im saving for the end of the video series, which is tr(AB) = tr(BA). However, in the majority of cases where one of A or B is invertible, we can derive it using one of the other properties. But in any case, lets start the visualisation. At the end of the previous video, we have described a matrix as a vector field. Lets expand on that a little bit. If you are given a matrix A, then the vector field will be constructed by attaching every point with position vector x by the vector Ax. A little note on the illustration on screen here. Technically if this illustration really describes Ax, then the vector field would