# Thread: Help with elementary Linear Algebra proof?

1. ## 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.

2. 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 $\displaystyle T=cU,$ for some scalar $\displaystyle c.$ since $\displaystyle T \neq 0,$ there exists $\displaystyle v \in V$ such that $\displaystyle T(v) \neq 0.$ but we also have $\displaystyle T(v)=cU(v)=U(cv),$ which gives us:

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