# Prove that V is a vector space. V=T(m, n), the set of linear transformations T : R^{m} rightarrow R^{n}, together with the usual addition and scalar multiplication of functions.

Prove that V is a vector space. , the set of linear transformations , together with the usual addition and scalar multiplication of functions.
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Step 1
the set of linear transformations together with the usual addition and scalar multiplication of functions are clearly defined.
next we need to determine if the five conditions of Definion 7.1 are all met.
If $T\left(u\right)$ and $T\left(v\right)$ are a linear transformation if for all vectors u and v and ${R}^{m},$ then their sum will also be a linear transformation in ${R}^{m}$, that is, is closed under addition.
If $T\left(u\right)$ is a linear transformation if for all vectors u in ${R}^{m}$ and c is a scalar then i and $cT\left(u\right)$ will also be a linear transformation thus is also closed under scalar multiplication.
M If $T\left(0\right)=0$ is an identical zero transformation, then if $T\left(u\right)$ is in ${R}^{m}$ linear transformation, then for all $T\left(u\right)$ in ${R}^{m}$, so that $T\left(0\right)$ is the zero transformation.
If $T\left(u\right)$ linear transformation, it has its inverse linear transformation $-T\left(u\right)$ so that

Step 2
a) We are checking whether you are

Therefore, commutativity is valid.
b) Check