Exercise 4.5.5

Answers

It’s a $3$ -linear function. We have that when the second and the third rows are held fixed, the function would be

δ (\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix}) = (A_{11}, A_{12}, A_{13}) \cdot (A_{23} A_{32}, 0, 0),

a inner product function. So $δ$ is linear for the first row. Similarly we can write

δ (\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix}) = (A_{21}, A_{22}, A_{23}) \cdot (0, 0, A_{11} A_{32}) = (A_{31}, A_{32}, A_{33}) \cdot (0, A_{11} A_{23}, 0) .

So $δ$ is a $3$ linear function.

jephianlin

2011-06-27 00:00