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(Solved): Consider the linear system of equations [[1,-1],[\alpha ,\beta ],[\delta ,\epsi lon]][[x_(1)],[x_(2) ...
Consider the linear system of equations
[[1,-1],[\alpha ,\beta ],[\delta ,\epsi lon]][[x_(1)],[x_(2)]]=[[0],[\gamma ],[\phi ]]
where \alpha ,\beta ,\gamma ,\delta ,\epsi lon,\phi are parameters.
(i) Assume
\alpha =\beta =\epsi lon=\phi =1,\gamma =2.
Find the value of \delta such that this linear system of equations has a unique solution. Justify your
answer algebraically, even though you are encouraged to use geometric intuitions. (4 points)
(ii) Still assume
\alpha =\beta =\epsi lon=\phi =1,\gamma =2.
and \delta is an arbitrary parameter. Find the vector x that minimizes ||Ax-b||. Your solution will
clearly be a 2\times 1 vector whose entries depend on \delta . (3 points)
(iii) Bonus: Using MATLAB plot the geometric location of the solution you found in (ii) as \delta varies
from -\infty to \infty . What shape is this? Does it make intuitive sense to you? (3 points)
(iv) Now let
\alpha =2,\beta =-2,\delta =3,\epsi lon=-3,\gamma =\phi =0.
Determine the existence, uniqueness, and complete set of solutions. (3 points). Explain steps please.