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Direct sums - Vanderbilt University
is that a vector space V is a direct sum of W 1 and W 2 if and only if every element of V can be written uniquely as a sum w 1 + w 2 with w 1 2W 1 and w 2 2W 2. Example. Consider the plane P = f(x;y;z) 2R3: z = 0gin R3 with L = f(x;y;z) 2 R3: x = y = 0g. Then these are both subspaces of R3 as we saw above, and their
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Injective and surjective functions
math.vanderbilt.eduif the image of T is an n-dimensional subspace of the (n-dimensional) vector space W. But the only full-dimensional subspace of a nite-dimensional vector space is itself, so this happens if and only if the image is all of W, namely, if T is surjective. In particular, we will say that a linear transformation between vector spaces V and
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The Cobb–Douglas Production Function
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