Let A Rmx be a matrix such that its row are linearly independent (as vectors...

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Let A € Rmx¹ be a matrix such that its row are linearly independent (as vectors in R") and let b E R™. such that the set H = {x € R": Ax=b} Is nonempty. 1) Prove that H is a closed and convex subset of R". 2) Let a ER". By considering the optimization problem (7) (min|lx-al|² s. t. Ax = b and using the KKT theorem, prove that P₁(a) = a + A¹ (AA)-¹(b - Aa). Let A € Rmx¹ be a matrix such that its row are linearly independent (as vectors in R") and let b E R™. such that the set H = {x € R": Ax=b} Is nonempty. 1) Prove that H is a closed and convex subset of R". 2) Let a ER". By considering the optimization problem (7) (min|lx-al|² s. t. Ax = b and using the KKT theorem, prove that P₁(a) = a + A¹ (AA)-¹(b - Aa).