Let $F:E_3\longrightarrow E_3$ defined by $$ F(\mathbf v)=\mathbf v\times(\mathbf i+\mathbf j+\mathbf k). $$ Determine whether $F$ is injective, surjective or bijective.
My attempt. $F$ is not injective because if $\mathbf v = \mathbf 0$, then $F(\mathbf v)=\mathbf 0$. But also if $\mathbf v'\neq\mathbf 0$ and $\mathbf v'$ is parallel to $\mathbf i + \mathbf j +\mathbf k$, then $F(\mathbf v')=\mathbf 0$. Hence $\mathbf v\neq\mathbf v'$ but $F(\mathbf v)=F(\mathbf v')$. Is this proof correct? Moreover, how can I determine if $F$ is surjective?
Thank You
Yes, it is correct. And since $F$ is a linear map between two $3$-dimensional vector spaces which is not injective, it cannot be surjective.
You can also prove that $F$ is not surjective observeng that every vector in the range of $F$ is orthogonal to $\mathbf{i}+\mathbf{j}+\mathbf k$. So, for instance $\mathbf{i}+\mathbf{j}+\mathbf k$ itself doesn't belong to the range of $F$.