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

• Mar 30th 2011, 03:46 AM
LHS
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?
• Mar 30th 2011, 04:02 AM
FernandoRevilla
Hint

For all $\displaystyle x\in V$, $\displaystyle x$ can be expressed univocally in the form $\displaystyle x=\lambda_1v_1+\ldots+\lambda_nv_n\;(\lambda_i\in\ mathbb{K}$).
• Mar 30th 2011, 04:27 AM
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]?
• Mar 30th 2011, 06:14 AM
FernandoRevilla
Quote:

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 $\displaystyle T$. Besides, $\displaystyle T(x)=\sum_{i=1}^{n}\lambda_i w_i$ is a linear map as a consequence of well known properties about coordinates.
• Mar 30th 2011, 06:40 AM
LHS
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?