Intereting Posts

Does $e^Xe^Y = e^Ye^X$ iff $=0$ hold once we are sufficiently close to the identity of a Lie group?
Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$
If I have that $X$ is a random variable satisfying $0\leq X \leq 1$, how can I show that $P\left(X \geq \frac{E(X)}{2}\right) \geq \frac{E(X)}{2}$?
Existence of a simultaneous rational approximation of real numbers in (0,1)
Hausdorff measure of rectifiable curve equal to its length
Uniqueness of compact topology for a group
A question about Darboux functions
Why are analytic functions functions of $z$ and not of $\bar{z}$?
Asymptotic expression for sum of first n prime numbers?
Question involving chess master (combinatorics)
matrix derivative of gradients
Finding the number of red balls drawn before the first black ball is chosen
Tarski monster groups with more than one prime
Example: Operator with empty spectrum
Find the minimum number of links to remove from digraph to make it acyclic

I would like to know how to define a basis of the space of linear maps : $ \mathcal{L} ( E , F ) $.

$ E $ and $ F $ are two differents vector spaces.

I’m not looking for how building a basis of its equivalent space $ \mathcal{M}_n ( \mathbb{R} ) $, i know it.

Thank you very much.

- Question about Axler's proof that every linear operator has an eigenvalue
- Prove that invertible metrices set is an open set in a given space, and the determinant is continuous
- Polar decomposition of real matrices
- Determining whether a symmetric matrix is positive-definite (algorithm)
- Calculating determinant with different numbers on diagonal and x everywhere else
- Does Nakayama Lemma imply Cayley-Hamilton Theorem?
- Question on Smith normal form and isomorphism
- A square matrix has the same minimal polynomial over its base field as it has over an extension field
- What does the sign $\propto$ mean?
- Geometric intuition of adjoint

If we have a basis for $E$ and a basis for $F$, we can use them to produce a basis for $\mathcal{L}(E,F)$ as follows. Let $e_1,\ldots,e_n$ be a basis for $E$ and $f_1,\ldots,f_m$ a basis for $F$.

For $i\in\{1,\ldots,n\}$ and $j\in\{1,\ldots,m\}$, define the linear map $\varphi_{ij}:E\to F$ by $\varphi_{ij}\left(a_1e_1+\cdots+a_ne_n\right) = a_if_j$. Then these $mn$ linear maps $\varphi_{ij}$ form a basis for $\mathcal{L}(E,F)$.

Are $E$ and $F$ finite dimensional? If so, here you go.

The second paragraph here should be helpful, too.

- Help with calculating infinite sum $\sum_{n=0}^{\infty}\frac1{1+n^2}$
- General condition that Riemann and Lebesgue integrals are the same
- Irreducible factors of $X^p-1$ in $(\mathbb{Z}/q \mathbb{Z})$
- Number of ways to connect sets of $k$ dots in a perfect $n$-gon
- How discontinuous can a derivative be?
- Proving : Every infinite subset of countable set is countable
- How prove that $P_{2n}<(n+1)^2$
- Proving a limit involved in the Lagrangian inversion of $\frac{\log\sqrt{1+x}}{\sqrt{1+x}}$
- What is the best calculus book for my case?
- Irreducible elements in $\mathbb{Z} $
- Why do people all the time exploiting almost sure properties of a stochastic process as if they were sure properties?
- Undecidable conjectures
- Understanding the multiplication of fractions
- Sequences of integers with lower density 0 and upper density 1.
- If $f(x) = \sin \log_e (\frac{\sqrt{4-x^2}}{1-x})$ then find the range of this function.