# Thread: Counter example R(T+U) subset of or equal R(T) + R(U)

1. ## Counter example R(T+U) subset of or equal R(T) + R(U)

Hello everyone,
I need help plz

Find a Counter example for:
If T and U are linear transformations from V to W and W are vector space then R(T+U) subset of or equal R(T) + R(U).

Thank you

2. What is R?

3. R = Rank
Thank you

4. The rank of a transformation (or matrix) is a number. How can a number be a subset of two other numbers?

Perhaps you wanted to find a counter example of

$rank(T+U) \leq rank(T) + rank(U)$ ?

5. oops
R(T) = the range of the linear transformation T
sorry
Thanks

6. Originally Posted by Nona
Hello everyone,
I need help plz

Find a Counter example for:
If T and U are linear transformations from V to W and W are vector space then R(T+U) subset of or equal R(T) + R(U).

Thank you
There is no counterexample for this since it's true: $y=Tx+Ux \in R(T)+R(U)$. The other inclusion is false: take $T,U: V \rightarrow V$ with $T=id_V$ and $U=-id_V$ then $R(T+U)= \{0 \}$ and $R(T)+R(U)=V$