Linear algebra proof (matrices and linear transformations)

[Ryker]

Homework Statement

Let T $$\in$$ L(V, W), where dim(V) = m and dim(W) = n. Let {v₁, ..., v_m} be a basis of V and {w₁, ..., w_n} a basis for W. Define the matrix A of T with respect to the pair of bases {v_i} and {w_j} to be the n-by-m matrix A = (a_ij), where

$$T(v_{i}) = \displaystyle\sum_{j=1}^{n}a_{ji}w_{j}, 1 \le i \le m, 1 \le j \le n.$$

The vector spaces V and W are isomorphic via the bases {v_i} and {w_j} to the spaces F_m and F_n, respectively. Show that ifx $$\in$$ F_m is the column vector corresponding to the vector x $$\in$$ V via the isomorphism, then Ax is the column vector in F_n corresponding to T(x). In other words, the correspondence between linear transformations and matrices is such that the action of T on a vectorx is realized by the matrix multiplication Ax.

Homework Equations

The Attempt at a Solution

I can see this is true, and when we learned this in class, it was pretty clear to me. But now I'm going through some linear algebra book and the stuff is introduced in a different way that confuses the hell out of me.

What I've tried to do here is just show what T(x) and Ax get you, and that they are equal. But I just can't get to that, it seems.

For T(x), I get the following:

$$T(x) = T(\displaystyle\sum_{i=1}^{m}b_{i}v_{i}) = \displaystyle\sum_{i=1}^{m}b_{i} \displaystyle\sum_{j=1}^{n}a_{ji}w_{i},$$

and for Ax:


$$Ax = \displaystyle\sum_{j=1}^{n} (\displaystyle\sum_{i=1}^{m}a_{ji}x_{i}) = \displaystyle\sum_{j=1}^{n} (\displaystyle\sum_{i=1}^{m}a_{ji}b_{i}v_{i}).$$

I don't know how to make the jump from v_i to w_j. Or have I gone completely in the wrong direction? Any help would be greatly appreciated.

 

[losiu99]

I believe there is a problem with matrix multiplication in your solution. The first warning: how did you get $$v_i$$'s into the result? It clearly shouldn't have happened, since the dimensions don't match. I'd say
$$Ax=\sum_{j=1}^n\left(\sum_{i=1}^m a_{ji}b_i w_j\right)$$
because that transformed coordinates are for this other basis, not V's basis.
Good luck!

 

[Ryker]

Hmm, I got v_i in my result, because I figured if x ϵ V, then x_i = b₁v₁ + b₂v₂ + ... + b_mv_m. So that's where I get stuck, I don't know how to get to having it expressed in terms of w_j, because A is defined in terms of a_ij and x as a linear combination of the basis vectors {v_i}.

How did you get to that solution you offered? If I got to that, then clearly I'd be good to go, since it would match T(x), but I just can't get to that.