Let be an m x n matrix. Given has nontrivial soln's, in each of the following demonstrate an example if you say yes of and or if you say no explain why.
Does in such that has:
a.) no solution
b.) a unique solution
c.) an infinite # of solutions
Let be an m x n matrix. Given has nontrivial soln's, in each of the following demonstrate an example if you say yes of and or if you say no explain why.
Does in such that has:
a.) no solution
b.) a unique solution
c.) an infinite # of solutions
Well, we know m can or cannot equal n.
For a, there will exist a vector when m < n I believe since we can have a pivot in the augmented column.
For b, I'm not sure.
For c, an infinite # of solutions means m < n (more columns than rows, because we'll have a pivot. But how can we answer definitely for a-c if we don't know what m is.
If is a solution to and is a non-trivial solution to then . Furthermore, since is non-trivial. Therefore, it is not possible to have unique solutions to . Therefore (b) is always false.
Define by .
The dimension of the space of all vectors such that is the rank of .
The dimension of all vectors such that is the nullity of .
By rank-nullity theorem we have .
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Case :
Since by assumption it means therefore there is such that is not solvable. Therefore (a) is true.
If we can show that there is a so that is solvable (say ) then is a solution (where is non-trivial which solves ) where . To show that there is such a we need to show that and it is sufficient to prove . But if this true? Well, almost. If is a matrix with zero entries then , and this is the only exception. Otherwise, it is true.
Try doing other cases.