[Math] There Exists a Linear Transformation that maps basis of Vector spaces of the same finite dimension.

Question

Suppose that V and W are vector spaces with the same dimension.

How we can show that there always exists a linear function that maps a basis of V to a basis of W?

Answer 1

Let $A=\{e_1,e_2...e_n\}$ be a basis of $V$ and $B=\{v_1,v_2...v_n\}$ a basis of $W$

Let $x \in V$, then $x=a_1e_1+...+a_ne_n$

Define $T: V \rightarrow W$ such that :

$T(x)=a_1v_1+...+a_nv_n$

We have the forall $i \in \{1,2...n\}$:

$e_i=0e_1+0e_2+...+0e_i+...+0e_n$, thus $T(e_i)=vi$

Also $T$ is linear:

If $x,y \in V$ then $$x=x_1e_1+...+x_ne_n$$ $$y=y_1e_1+...+y_ne_n$$

Thus $$T(x+y)=T((x_1+y_1)e_1+...+(x_n+y_n)e_n)=(x_1+y_1)v_1+...+(x_n+y_n)v_n=(x_1v_1+...+x_nv_n)+(y_1v_1+...+y_nv_n)=T(x)+T(y)$$

I leave to you the proof of the claim: $$T(ax)=aT(x), \forall a \in \mathbb{F}$$ where $\mathbb{F}$ is the field.