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

• Dec 2nd 2009, 04:43 PM
Nona
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
• Dec 4th 2009, 04:00 AM
Defunkt
What is R?
• Dec 4th 2009, 10:42 AM
Nona
R = Rank
Thank you
• Dec 4th 2009, 11:00 AM
Defunkt
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

$\displaystyle rank(T+U) \leq rank(T) + rank(U)$ ?
• Dec 4th 2009, 11:59 AM
Nona
oops
R(T) = the range of the linear transformation T
sorry
Thanks
• Dec 4th 2009, 02:25 PM
Jose27
Quote:

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: $\displaystyle y=Tx+Ux \in R(T)+R(U)$. The other inclusion is false: take $\displaystyle T,U: V \rightarrow V$ with $\displaystyle T=id_V$ and $\displaystyle U=-id_V$ then $\displaystyle R(T+U)= \{0 \}$ and $\displaystyle R(T)+R(U)=V$