- #1
JD_PM
- 1,131
- 158
- TL;DR Summary
- I am wondering if there is a theorem to check whether a linear map is surjective or not in function of the value of the parameters.
I am given the following matrix representation of a linear mapping
$$L_{\alpha}^{\beta}=
\begin{pmatrix}
1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 & 0 & 1 \\
0 & a & 0 & b & 0 & c \\
a & 0 & b & 0 & c & 0 \\
0 & a^3 & 0 & b^3 & 0 & c^3 \\
a^3 & 0 & b^3 & 0 & c^3 & 0 \\
\end{pmatrix}
$$
Where ##a,b,c \in \Re##
And I am asked to find the values that make L surjective.
Well, if the question was about what values of ##a,b,c## make ##L## injective I'd go for the rank-nullity theorem and see for what values I get that the nullity of ##L## is zero (as there's a theorem stating that if the nullity of ##L = 0 \rightarrow L## is injective).
My question is: is there an analogous theorem to check surjectivity?
I am aware of the basic definition of a surjective map: L is surjective if for every element ##y## in the codomain ##Y## there is at least one element ##x## in the domain ##X##.
$$L_{\alpha}^{\beta}=
\begin{pmatrix}
1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 & 0 & 1 \\
0 & a & 0 & b & 0 & c \\
a & 0 & b & 0 & c & 0 \\
0 & a^3 & 0 & b^3 & 0 & c^3 \\
a^3 & 0 & b^3 & 0 & c^3 & 0 \\
\end{pmatrix}
$$
Where ##a,b,c \in \Re##
And I am asked to find the values that make L surjective.
Well, if the question was about what values of ##a,b,c## make ##L## injective I'd go for the rank-nullity theorem and see for what values I get that the nullity of ##L## is zero (as there's a theorem stating that if the nullity of ##L = 0 \rightarrow L## is injective).
My question is: is there an analogous theorem to check surjectivity?
I am aware of the basic definition of a surjective map: L is surjective if for every element ##y## in the codomain ##Y## there is at least one element ##x## in the domain ##X##.