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Unformatted text preview: 1. Deﬁne the range of a matrix A. R(A) = {v  Ax = v for some vector x}. In words: The range of A is the set of all vectors v for which the equation Ax = v has a solution. Equivalent deﬁnition: Range of A is the span of columns of A. 2. Deﬁne the null space of A. N (A) = {v  Av = 0}. In words: The null space of A is the set of all the solutions to the equation Av = 0. 3. Explain why the range and the null space of A are vector subspaces. We need to check that 0, sums and constant multiples are in the range and in the null space. For R(A): A0 = 0 therefore 0 ∈ R(A). If v ∈ R(A) then v = Ax for some vector x. Also, A(cx) = c(Ax) = cv therefore any multiple of v is also in the range of A. If v, w ∈ R(A) then v = Ax and w = Ay for some vectors x, y . Then A(x + y ) = Ax + Ay = v + w. Thus v + w ∈ R(A). Thus R(A) is a vector subspace. For N (A): A0 = 0 therefore 0 ∈ N (A). If v ∈ N (A) then Av = 0. Also, A(cv ) = cAv = c0 = 0. Therefore any multiple of v is also in the null space of A. If v, w ∈ N (A) then Av = 0 and Aw = 0. Then A(x + y ) = Ax + Ay = 0 + 0 = 0. Thus v + w ∈ N (A). Thus N (A) is a vector subspace. 1 ...
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 Spring '10
 Dr.Dmitry

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