2
$\begingroup$

Suppose we have linear transformations $T : V \to W$ for finite dimensional vector spaces $V$ and $W$.
We can certainly think of $\mathcal{L}(V,W)$ as a vector space of all those transformations (provided addition and multiplication by scalar of transformations are defined and satisfy certain axioms).

While a vector space doesn't really need a basis and a dimension to exist, is there any meaning for a basis and dimension of this vector space $\mathcal{L}(V,W)$? Suppose $V$ is $n$-dimensional and $W$ is $m$-dimensional, what would be the dimension of $\mathcal{L}(V,W)$?

I was trying to make an analogy for the matrix representation of a linear transformation $T:V \to W$ as simply an isomorphism from $\mathcal{L}(V,W)$ to $F^{n\times m}$, the vector space of all $n \times m$ matrices over the field $F$.

Any clue about those points ?

$\endgroup$
  • 1
    $\begingroup$ A good reference for this is Section 2.4 from Linear Algebra by Friedberg et al. $\endgroup$ – user70962 Apr 24 '13 at 20:06
2
$\begingroup$

In order:

Yes, there is meaning for basis and dimension for $L(V,W)$, and they have meaning for every other vector space for that matter.

The dimension is $\dim(V)*\dim(W)$.

Yes, you can show that $L(V,W)\cong M_{\dim(V),\dim(W)}(\Bbb F)$ in such a way that evaluation of these transformations corresponds to matrix multiplication.

$\endgroup$
1
$\begingroup$
  1. You are Given Two Vector Spaces V and W.

    Now Any finite Dimensional Vector space Is Defined By its basis set B and Its field F.

2.Hence V=d(A,F) and W=d(B,F) where A is the Basis set Of V and B is that of W and F is the common field Of the vector Space.(d represents that it is defined by the following Sets)

  1. Hence We have Fixed A, B ,And F to define V(m)and W(n).

4. We know that Any Linear Transformation between Two fixed Basis of V and W respectively can be uniquely represented by A Matrix(mXn). Also, ANY MATRIX IN F(mXn) REPRESENTS AN UNIQUE LINEAR TRANSFORMATION FROM V->W AS THE BASES ARE FIXED.

  1. HENCE, the Required vector space is just the set of all F(mXn). and its dimension is m*n.

Beautiful isn't it!!!!!!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.