Question

x1-5x2+x3+3x4=1

2x1-x2-3x3-x4=3

-3x1-3x3+7x3+5x4=k

1 ) There is exactly one real number k for which the system has at least one solution; determine this k and describe all solutions to the resulting system.

2 ) Do the solutions you found in the previous part form a linear subspace of R4?
3 ) Recall that a least squares solution to the system of equations Ax = b is a vector x minimizing the length |Ax=b| suppose that

{x1,x2,x3,x4} = {2,1,1,1}

is a least squares solution to the above system of equations. Determine k.

Answer 1