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Numerical Linear Algebra with Applications by William Ford

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16.7 Problems

16.1 Three variables x, y, and z are required to satisfy the equations

3xy+7z=02xy+4z=1/2xy+z=16x4y+10z=3

si195_e

Is there a solution in the normal sense, not a least-squares solution? If so, is it unique, or are there infinitely many solutions?

16.2 If A is an invertible matrix and assuming exact arithmetic, are the solutions using Gaussian elimination and least squares identical? Prove your assertion.

16.3 If A is an m × n matrix, mn, and b is an m × 1 vector, show that if ATb = 0, b is orthogonal to the range of A.

16.4 If A is an m × n full-rank matrix, mn, show that

κ(A)=||A||2||A||2=σ1σn,

where σ1 and σn are the largest ...

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