Intereting Posts

Hilbert class field of $\mathbb Q(\sqrt{-39})$
Prove that CX and CY are perpendicular
How many ways so that $x+y+z\leq n$?
What pattern does set theory study?
Show that $\lim\limits_{y\downarrow 0} y\mathbb{E}=0$.
Matrix representation of the adjoint of an operator, the same as the complex conjugate of the transpose of that operator?
How to show $p_n$ $\leq$ $2^{2^n}$?
Combinatorics Pigeonhole problem
Geodesics on a polyhedron
Nilpotent matrix and basis for $F^n$
Analytic $F(z)$ has $f(z)$ as derivative $\implies$ $\int_\gamma f(z)\ dz = 0$ for $\gamma$ a closed curve
Calc the sum of $\sum_{k = 0}^{\infty} \frac{(-1)^k}{k} \sin(2k)$
Why people use the Gram-Schmidt process instead of just chosing the standard basis
$R$ is a commutative integral ring, $R$ is a principal ideal domain imply $R$ is a field
Sampling $Q$ uniformly where $Q^TQ=I$

If $M$ and $ N$ be the subspaces of a vector space $V$ then prove that $(\dim M) + (\dim N) = (\dim M+N) + (\dim M \cap N)$.

- How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?
- When does $V = \ker(f) \oplus \operatorname{im}(f)$?
- Effect the zero vector has on the dimension of affine hulls and linear hulls
- What is the significance of reversing the polarity of the negative eigenvalues of a symmetric matrix?
- Characterisation of inner products preserved by an automorphism
- Hermitian Matrices are Diagonalizable
- Solutions to the matrix equation $\mathbf{AB-BA=I}$ over general fields
- Inverse of a matrix with uniform off diagonals
- Prove that, if $\{u,v,w\}$ is a basis for a vector space $V$, then so is $\{u+v, v+w, u+v+w\}$.
- Set of all unitary matrices - compactness and connectedness.

Let $u_1, \ldots , u_k$ be basis of $M\cap N$,

$v_1, \ldots v_s, u_1, \ldots , u_k$ – basis of $M$,

$u_1, \ldots , u_k, w_1, \ldots w_t$ – basis of $N$.

It is easy to see that $v_1, \ldots v_s, u_1, \ldots , u_k, w_1, \ldots w_t $ is basis of $M+N$

- Prove that $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ for all real $x$ and $y$.
- Curve in $\mathbb{A}^3$ that cannot be defined by 2 equations
- Baker-Hausdorff Lemma from Sakurai's book
- Are there real-life relations which are symmetric and reflexive but not transitive?
- find maximum area
- Let $\{K_i\}_{i=1}^{\infty}$ a decreasing sequence of compact and non-empty sets on $\mathbb{R}^n.$ Then $\cap_{i = 1}^{\infty} K_i \neq \emptyset.$
- How to solve $\ x^2-19\lfloor x\rfloor+88=0 $
- How could we define the factorial of a matrix?
- what is the mutual information of three variables?
- The determinant of adjugate matrix
- Finding simple, step, and continuous functions to satisfy Lebesgue integral conditions
- How to evaluate this improper integral $\int_{0}^{\infty}\frac{1-x}{1-x^{n}}\,dx$?
- Inducing homomorphisms on localizations of rings/modules
- When does a polynomial divide $x^k – 1$ for some $k$?
- Metric space and continuous function