# Let left{v_{1}, v_{2}, dots, v_{n}right} be a basis for a vector space V.

Let be a basis for a vector space V. Prove that if a linear transformation satisfies then T is the zero transformation.
Getting Started: To prove that T is the zero transformation, you need to show that $T\left(v\right)=0$ for every vector v in V.
(i) Let v be an arbitrary vector in V such that
(ii) Use the definition and properties of linear transformations to rewrite as a linear combination of .
(iii) Use the fact that to conclude that , making T the zero tranformation.

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Step 1
Let be the basis for a vector space.
Define the map,
To prove, the above defined map is a linear transformation.
Step 2

$\text{Also}$

Therefore, the above defined map is a linear transformation and this is true for every vector in V.
Step 3
Let v be an arbitrary vector in V such that
Where, ${C}_{i}$ 's are scalars.
The vector v can be written as such linear combination since is the basis for a vector space V.
Apply the transformation T on both sides and use the fact that T is linear transformation.