a) The matrix A that has V as it's row space is equals to A = [ 1 1 1 ; 2 1 0 ]
b) The matrix B that has V as it's null space is equals to B = [ b₁ -2b₁ b₁ ]
Let V is the subspace spanned by (1, 1, 1) and
(2,1, 0),
a) Here, Row space(A) = V = span{(1, 1, 1),(2,1, 0)}
Since, row space is spanned by row vectors of A .
Thus, A = [ 1 1 1 ; 2 1 0 ] where (1,1,1) first row.
b) Given that nullspace(B) = V
= span{ (1,1,1)' , (2,1,0)'}
where (1,1,1)' is transpose of (1,1,1) vector. Since, dim (nullspace B ) = 2.
Rank of B = number of columns - dim(null space)
=> Rank of B = 3 -2 = 1
let B = [ b₁ b₂ b₃ ], B (1,1,1)' = 0
=> [ b₁ b₂ b₃ ] [1,1,1]' = 0 => b₁ + b₂ + b₃ = 0 --(1)
and B (2,1,0)' = 0 => 2b₁ +b₂ = 0 --(2)
from (1) and (2) we get , b₂ = -2b₁ and b₃ = b₁
So, B = [ b₁ -2b₁ b₁ ]
Hence, we get the matrix A and matrix B .
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