Let A Rmxn with m > n and A = [10uv be a singular value...

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Let A € Rmxn with m > n and A = [10₁u₂v be a singular value decomposition of A. Assume that σι > ... Σση > 0. 1. (7 points) Consider A = 1(1/0₁) viu. Prove that for any y € Rm, AA¹y is the projection of y to the subspace span (u₁,..., un) CRm. (You can use the definition in page 5, Lecture 19.) 2. (7 points) Additionally given b = Rm, we want to find € R" such that || Ax – b|| is minimized. (Recall that ||v||2 Ei vis its Euclidean length.) Prove that x = A¹b is an optimal solution. (Hint: Geometrically, given A and b, what should Ar be in order to minimize || Axb||²?) = Let A € Rmxn with m > n and A = [10₁u₂v be a singular value decomposition of A. Assume that σι > ... Σση > 0. 1. (7 points) Consider A = 1(1/0₁) viu. Prove that for any y € Rm, AA¹y is the projection of y to the subspace span (u₁,..., un) CRm. (You can use the definition in page 5, Lecture 19.) 2. (7 points) Additionally given b = Rm, we want to find € R" such that || Ax – b|| is minimized. (Recall that ||v||2 Ei vis its Euclidean length.) Prove that x = A¹b is an optimal solution. (Hint: Geometrically, given A and b, what should Ar be in order to minimize || Axb||²?) =

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ANSWER Q1 To prove that AAty is the projection of y onto the subspace spanned by u u we need to show that it satisfies the properties of a projection First lets express y as a linear combination of u ... View the full answer