How can I solve this problem?
Let Z be a 4 x 5 matrix. A basis for the null space of Z is {ππ} where ππ β π. If the system of equations (Z^T)π± = π¦ is consistent, how many solutions does the system have? Justify your answer. *Note: Z^T is Z transpose.
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$\begingroup$Try to find first what is the nulity of z^t, and here you dont need actually field also i am assuming it is complex then the solution will be cardinality of field ^n(z^t).
$\endgroup$ 1 $\begingroup$At all cases THE NUMBER OF SOLUTIONS ARE EQUAL TO THE NUMBER OF INFORMATION PROVIDED and whenever they are equal the system is regarded as being solvable, relate this to your question
Check the number of information, condition on the system to conclude
$\endgroup$ $\begingroup$Let Z be m * n matrix. From the information provided it is clear that the dimension of Null Space is 1. And we know that n - r = dim(Null Space). So r = 5-1 = 4. So, the rank of the matrix is 4.
As the matrix is full row rank(r = m) . You can solve it for any right hand side. Do Gaussian Elimination. You will see there is no zero column as the matrix is full row rank. So, there is no condition that is needed to satisfy on the right hand side ( like zero row value must correspond to zero value on right side). So the number of solution is infinte.
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