Solution to linear system of equationsNotation. Let y, a, and b be n × 1, p × 1, and q × 1 real vectors. Let also X and Z be n × p and n × q real matrices.Suppose that there is no solution, a, to y = X a.Question. What are the conditions on Z such that y = X a + Z b has no solution for each choice of b?Context. I came across this problem in the context of linear regression. The fact that y = X a has no solution can be interpreted as "no hyperplane can perfectly fit the data". I am analysing an extension of this problem which has lead me to the need of finding something similar for " y = X a + Z b has no solution".

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Solution to linear system of equationsNotation. Let   y,   a, and   b be   n  ×  1,   p  ×  1, and   q  ×  1 real vectors. Let also   X and   Z be   n  ×  p and   n  ×  q real matrices.Suppose that there is no solution,   a, to   y  =  X  a.Question. What are the conditions on   Z such that   y  =  X  a  +  Z  b has no solution for each choice of   b?Context. I came across this problem in the context of linear regression. The fact that   y  =  X  a has no solution can be interpreted as &quot;no hyperplane can perfectly fit the data&quot;. I am analysing an extension of this problem which has lead me to the need of finding something similar for &quot;  y  =  X  a  +  Z  b has no solution&quot;.

Lamont Adkins · Accepted Answer

You are essentially asking, when does   y  =      [    X         Z    ]    b have no solution   b? Or in other words, when is   y not in the direct sum of the column spaces of   X and   Z?An obvious necessary, but not sufficient, condition on   Z is that   y  =  Z  u has no solution   u. I don&#039;t think there&#039;s a nice complete set of conditions on   Z; in practice you can row-reduce   [  X     Z  ] and see if   y  =  [  X     Z  ]  b is a consistent linear system, or equivalently, solve the least-squares problem      [                                        X            T                    X                                      X            T                    Z                                                  Z            T                    X                                      Z            T                    Z                      ]    b  =      [                                        X            T                                                            Z            T                                ]    yfor   b and see if the residual is zero.

Solution to linear system of equations Notation. Let y , a , and b be n &#x0

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