Exercise 6.2.12

Answers

If $x \in {(R (L_{A^{∗}}))}^{⊥}$ , this means that $x$ is orthogonal to $A * y$ for all $y \in 𝔽^{m}$ . So we have

0 = ⟨ x, A^{∗} y ⟩ = ⟨ Ax, y ⟩

for all $y$ and hence $Ax = 0$ and $x \in N (L_{A})$ . Conversely, if $x \in N (L_{A})$ , we have $Ax = 0$ . And

⟨ x, A * y ⟩ = ⟨ Ax, y ⟩ = 0

for all $y$ . So $x$ is a element in ${(R (L_{A^{∗}}))}^{⊥}$ .

jephianlin

2011-06-27 00:00