# Help with elementary Linear Algebra proof?

• Jun 7th 2009, 01:40 PM
paupsers
Help with elementary Linear Algebra proof?
I've been brushing up on some linear algebra this summer, and here's a problem I found in a book that I can't seem to solve. Any help is appreciated!

Let V and W be vector spaces, and let T and U be nonzero linear transformations from V into W. If the intersection of R(T) and R(U)* is 0, prove that {T, U} is a linearly independent subset of L(V, W).**

*R(T) and R(U) denote the range of the linear transformation of T and U.
** L(V, W) denotes the space of linear transformations from V to W.
• Jun 7th 2009, 01:53 PM
NonCommAlg
Quote:

Originally Posted by paupsers
I've been brushing up on some linear algebra this summer, and here's a problem I found in a book that I can't seem to solve. Any help is appreciated!

Let V and W be vector spaces, and let T and U be nonzero linear transformations from V into W. If the intersection of R(T) and R(U)* is 0, prove that {T, U} is a linearly independent subset of L(V, W).**

*R(T) and R(U) denote the range of the linear transformation of T and U.
** L(V, W) denotes the space of linear transformations from V to W.

suppose they are not linearly independent. then $T=cU,$ for some scalar $c.$ since $T \neq 0,$ there exists $v \in V$ such that $T(v) \neq 0.$ but we also have $T(v)=cU(v)=U(cv),$ which gives us:

$T(v) \in R(T) \cap R(U) = (0).$ contradiction!