Prove/Disprove: Sum of im/ker of linear transformation contained in ker/im of each linear trasnfromation

Question

Let $V$ be a finite vector space and let $T,S:V\to W$ linear transformations

Prove/Disprove the following:

1. If ${\rm Im}(T+S)\subseteq {\rm Im}(T)$ so ${\rm Im}(S)\subseteq {\rm Im}(T)$

2. If ${\rm Ker}(T+S)\subseteq {\rm Ker}(T)$ so ${\rm Ker}(S)\subseteq {\rm Ker}(T)$

1.Proof: Let $w\in {\rm Im}(S)$ therefore exists $v\in V$ s.t $S(v)=w$. Let take this $v\in V$ and check the image of in under $T+S$.

$$(T+S)v=_{(1)}T(v)+S(v)=T(v)+w$$

$(1)$=linearity of the transformation

Where $T(v)\in W$ and $w\in W$, but it is given that ${\rm Im}(T+S)\subseteq {\rm Im}(T)$ ,so both $T(v)+w\in {\rm Im}(T)$,but ${\rm Im}(T)$ is a subspace so $T(v)\in {\rm Im}(T)$ and $w\in {\rm Im}(T)$

2.Disprove: Let $T=I$ and $S=0$ so $I={\rm Ker}(T+S)\in {\rm Ker}(T)=I$ but $V={\rm Ker}(S)\not\subseteq {\rm Ker}(T)=\{0\}$

Are those proofs valid?

Answer 1

Yes, up to a couple of small errors:

• The equality $(1)$ holds by definition of $T+S$, not by linearity of $T$ and/or $S$
• At the end of the first argument you should write "so $T(v)+w \in \mathrm{Im}(T)$, but $\mathrm{Im}(T)$ is a subspace AND $T(v) \in \mathrm{Im}(T)$, THEREFORE $w=(T(v)+w)-T(v) \in \mathrm{Im}(T)$".
• In $2.$, I think you want to take $V=W\neq \{0\}$, $T=\mathrm{Id}$ and $\{0\}=\ker(T)=\ker(T+S)$, right?