Mike Artin suggested a neat higher-level proof of that dimension formula in 43. From all
Chapter 3, Problem 44(choose chapter or problem)
Mike Artin suggested a neat higher-level proof of that dimension formula in 43. From all inputs V in V and w in W, the "sum transformation" produces v+w. Those outputs fill the space V + W. The nullspace contains all pairs v = u, W = -u for vectors u in V n W. (Then v + W = u - u = 0.) So dim(V + W) + dim(V n W) equals dim(V) + dimeW) (input dimension/rom V and W) by the crucial formula dimension of outputs + dimension of nullspace = dimension of inputs. an m by n matrix of rank r, what are those 3 dimensions? Outputs = column space. This question will be answered in Section 3.6, can you do it now
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