Chapter 1.5 Inverses-LinearSystems

Proof have examples of three cases suppose c1 and c2

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Unformatted text preview: les of three cases. Suppose c1 and c2 are two distinct solutions of Ax = b. Then c0 = c1 − c2 = 0. We have Ac0 = A(c2 − c2 ) = Ac1 − Ac2 = b − b = 0. For any scalar λ, set c = c1 + λc0 . Ac = A(c1 + λc0 ) = Ac1 + λAc0 = b + 0 = b, so c is a solution of the system. Hence, there are infinitely many solutions to the system. Outline Number of Solutions Solving Multiple Systems More Results On the Inverse Properties of an Invertible Matrix Theorem If A is an n × n matrix, then the following are equivalen...
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