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Let A be an m × n matrix. Prove that every vector x in Rn
Chapter 6, Problem 23E(choose chapter or problem)
Let A be an m × n matrix. Prove that every vector x in Rn can be written in the form x = p + u, where p is in Row A and u is in Nul A. Also, show that if the equation Ax = b is consistent, then there is a unique p in Row A such that Ap = b.
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QUESTION:
Let A be an m × n matrix. Prove that every vector x in Rn can be written in the form x = p + u, where p is in Row A and u is in Nul A. Also, show that if the equation Ax = b is consistent, then there is a unique p in Row A such that Ap = b.
ANSWER:Solution 23EConsider an matrix By the orthogonal decomposition theorem, for the vector subspace of , each y in can be written uniquely as y=+ z. Here , .So each x in can be written as x=p+u, with P in Row A and u in As the orthogonal complement of the row space of A i