Define subspaces $U_1 \leq V$ and $U_2 \leq V$, can I then claim $\dim(U_1+U_2)=\dim V$

linear algebra

Suppose the following:

  • ($v_1,…,v_m$) is a basis of $V$
  • ($w_1,…,w_n$) is a basis of $W$
  • $T \in L(V,W)$

If I were to define two subspaces $U_1:=\{\forall v\in V | Tv \in \text{span}(w_1,…,w_k)$, with $1\leq k < n\}$ and $U_2:= \{ \forall v\in V | Tv \in \text{span}(w_{k+1},…,w_n) \}$.

Would it be possible for me to conclude form here that: $\dim(U_1+U_2)=\dim V=m$? I think should be able to make this claim, but I am not quite sure how to show it.

Best Answer

This is false.

Consider $T: \Bbb R^2 \rightarrow \Bbb R^2$ such that $e_1 \mapsto e_1 + e_2$ and $e_2 \mapsto e_1 + e_2$. Now let $w_1 = e_1$ and $w_2 = e_2$, $k=1$. You will find $U_1 = U_2 = 0$.

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