By computing both sides, show that for an m × n matrix A, vectors u and...

By computing both sides, show that for an m × n matrix A, vectors u and v ∈ Rn , and a scalar s ∈ R, we have (a) A(sv) = s(Av); (b) A(u + v) = Au + Av; (c) A(0) = 0. Here 0 denotes the zero vector. Is the meaning of 0 on the two sides identical? Why or why not? Hint: Let x = (x1, . . . , xn) and y = (y1, . . . , yn) be vectors in Rn . The definition of vector equality is that x = y if and only if, for each i between 1 and n, we have xi = yi . So verify this property for the vectors in (a), (b), (c).

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Colby Messinger answered 1 year ago

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