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Damon Stokes 2022-06-27 Answered
For each m , n N think R m as the space of real column vectors of size m and R m × n as the space of matrices of size m × n.
Let d N
Let a : { 1 , , 2 d } { 1 } × { 0 , 1 } d be an enumeration (injective and surjective map).
Let A R ( d + 1 ) × 2 d be the matrix whose columns are a ( 1 ) , , a ( 2 d )
Is it true that for each b { 1 } × [ 0 , 1 ] d there exists x [ 0 , + ) 2 d such that A x = b ??
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Answers (1)

Carmelo Payne
Answered 2022-06-28 Author has 25 answers
Yes, there is an x with x i [ 0 , 1 ].
Essentially, you want to solve
n = 1 2 d x n = 1 , n 2 k 1 ( mod 2 ) x n = b k
and you can see
x n = k = 1 d f k ( n ) , f k ( n ) := { b k n 2 k 1 ( mod 2 ) 1 b k n 2 k 0 ( mod 2 )
is a solution.
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