Intereting Posts

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Assuming: $\forall x \in :f(x) > x$ Prove: $\forall x \in :f(x) > x + \varepsilon $
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real meaning of divergence and its mathematical intuition
Eigenvalues For the Laplacian Operator

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.

- Is Singular value decomposition suitable for solving matrix equations Ax=b?
- The kernel and range of the powers of a self-adjoint operator
- Prove that if $(v_1,\ldots,v_n)$ spans $V$, then so does the list $(v_1-v_2,v_2-v_3,\ldots,v_{n-1}-v_n,v_n).$
- What are the technical reasons that we must define vectors as “arrows” and carefully distinguish them from a point?
- Why is the determinant of a symplectic matrix 1?
- Help with Cramer's rule and barycentric coordinates
- If $A$ and $B$ are positive-definite matrices, is $AB$ positive-definite?
- Change of Basis Confusion
- Find trace of linear operator
- Matrix is conjugate to its own transpose

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.

- Yoneda's lemma and $K$-theory.
- Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
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- Computing the dimension of a vector space of matrices that commute with a given matrix B,
- Description of $\mathrm{Ext}^1(R/I,R/J)$
- can we use generating functions to solve the recurrence relation $a_n = a_{n-1} + a_{n-2}$, $a_1=1$, $a_2=2$?
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- Does Chaitin's constant have infinitely many prime prefixes?
- What are the coefficients of the polynomial inductively defined as $f_1=(x-2)^2\,\,\,;\,\,\,f_{n+1}=(f_n-2)^2$?