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Examples of Linear Maps

Linear Algebra: Here are a few problems on linear maps. Part 1: Are the following maps L:R^3 to R^3 linear? (a) L(x, y, z) = (x+1, x-y-2, y-z), (b) L(x, y, z) = (x + 2y, x-y-2z, 0). Part 2: Suppose L:R^3 to R^2 is linear and defined on the standard basis by L(e1) = (1, 2), L(e2) = (0, 3), and L(e3) = (1, -1). Compute L(2,-1,3). For both parts, we explain linearity in terms of linear combinations and in terms of matrix-based maps.
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