# Thread: How to prove there is a unique linear map V->W for elements of the basis of V

1. ## How to prove there is a unique linear map V->W for elements of the basis of V

If I have a basis of V, say v1,..,vn and w1,...,wn elements of W, how can I show there is a unique transformation T(vi)=wi for all i 1..n?

2. Hint

For all $x\in V$, $x$ can be expressed univocally in the form $x=\lambda_1v_1+\ldots+\lambda_nv_n\;(\lambda_i\in\ mathbb{K}$).

3. So can I let T(x)= lambda[1]*w[1] + ... + lambda[n]*w[n], for some x element of V, then say the right hand side is uniquely determined due to lambda[i] being uniquely determined and as you can write x in terms of lambda[i]*v[i] there has to be a unique map between each v[i] to w[i]?

4. Originally Posted by LHS
So can I let T(x)= lambda[1]*w[1] + ... + lambda[n]*w[n], for some x element of V, then say the right hand side is uniquely determined due to lambda[i] being uniquely determined and as you can write x in terms of lambda[i]*v[i] there has to be a unique map between each v[i] to w[i]?

Right, that proves the unicity of $T$. Besides, $T(x)=\sum_{i=1}^{n}\lambda_i w_i$ is a linear map as a consequence of well known properties about coordinates.

5. Ah yes, so you can show it's a linear map, then let x=v[i] and you can show that T(v[i])= 0*w[1]+...+1*w[i]+...+0*w[n], then go about showing T is unique?