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# s1 - CO350 Assignment 1 Solutions Exercise 1(linear algebra...

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CO350 Assignment 1 Solutions Exercise 1: (linear algebra review.) Let A R m × n and b R m . (a) If the system Ax = b has a solution and the columns of A are not linearly indepen- dent, then Ax = b has inﬁnitely many solutions. Proof. Suppose that x 0 R n satisﬁes Ax 0 = b . If the columns of A are not linearly independent, then there is a non-zero vector x 1 R n such that Ax 1 = 0. Now, for each a R , we have A ( x 0 + ax 1 ) = Ax 0 + aAx 1 = b . Hence Ax = b has inﬁnitely many solutions. (b) Show that, if the system Ax = b has a solution and the columns of A are linearly independent, then Ax = b has a unique solution. Proof. We will prove the equivalent statement that, if Ax = b has two distinct solutions, then the columns of A are not linearly independent. Suppose that x 1 R n and x 2 R n are distinct solutions of Ax = b . Then x = x 2 - x 1 is a non-zero vector and Ax = Ax 2 - Ax 1 = 0. Hence, the columns of A are not linearly independent. Exercise 2:

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s1 - CO350 Assignment 1 Solutions Exercise 1(linear algebra...

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