Intereting Posts

(Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$
Is this a perfect set?
How find this $a_{1}+a_{2}+\cdots+a_{500}=b_{1}+b_{2}+\cdots+b_{500}$?
Contractibility of the space of sections of a fiber bundle
Expectation of Minimum of $n$ i.i.d. uniform random variables.
Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?
Why does the boundary of the Mandelbrot set contain a cardioid?
Does one of $L^\infty$ and $L^p, p \in (0, \infty)$ contain the other?
Isomorphisms Between a Finite-Dimensional Vector Space and its Dual
A question on morphisms of fields
Expansion of this function in a Fourier-Legendre series
What are the symmetries of a colored rubiks cube?
Find numbers in a set whose sum equals x
Why is the Cauchy product of two convergent (but not absolutely) series either convergent or indeterminate (but does not converge to infinity)?
Expected number of steps till a random walk hits a or -b.

Let $H$ be:

$$H=\{(x, y, z, w)^T : x^2-y^2+3w^3=0\}$$

I know it has to be closed under addition and scalar multiplication, and it does contain the 0 vector. But how do I prove it’s closed under the first two? Do I just keep finding examples for what $x, y, z$, and $w$ could be and just keep scaling them and adding them? There’s gotta be a more efficient way than that. Help please?

- Eigenvector of unitary matrix
- Is the absolute value function a linear function?
- Show that if $T_1$, $T_2$ are normal operators that commute then $T_1+T_2$ and $T_1T_2$ are normal.
- Find the number of $n$ by $n$ matrices conjugate to a diagonal matrix
- Sub-determinants of an orthogonal matrix
- A non-negative matrix has a non-negative inverse. What other properties does it have?

- Prove that if Rank$(A)=n$, then Rank$(AB)=$Rank$(B)$
- References on the History of Linear Algebra
- What is the best way to compute the pseudoinverse of a matrix?
- The number of subspaces of a vector space forming direct sum with a given subpace
- Proof that Every Positive Operator on V has a Unique Positive Square Root
- Dimension of vector space of matrices with zero row and column sum.
- Why Aren't “Similar” Matrices Actually the Same?
- Most important Linear Algebra theorems?
- Is there any matrix $2\times 2$ such that $A\neq I$ but $ A^3=I$
- Showing that $x^{\top}Ax$ is maximized at $\max \lambda(A)$ for symmetric $A$

Well, it is not a subspace. The way you can *know* beforehand (and if you prove the following then you will **always** know) is: when given as a function of the coordinates, a subset of $\;\Bbb R^n\;$ is a subspace iff it is a homogeneous equation in the $\;n\;$ coordinates of degree $\;1\;$, i.e. iff it is of the form

$$a_1x_1+…+a_nx_n=0\;,\;\;a_i\in\Bbb R$$

*Hint:* Is $(1,1,0)\in H$? is $(1,-1,0)\in H$? Is their sum $\in H$ (this assumes that $2\ne 0$; as a matter of fact, $H$ would be a subspace of $\mathbb F_2^4$)?

- Evaluation of a continued fraction
- Sum of power functions over a simplex
- A Ramanujan infinite series
- What does it mean for something to be true but not provable in peano arithmetic?
- Why formulate continuity in terms of pre-images instead of image?
- Laplace's Equation in Spherical Coordinates
- How does Dummit and Foote's abstract algebra text compare to others?
- Approximating Stirling's number of the second kind to allow for large inputs
- How to show that $\mathfrak{sl}_n(\mathbb{R})$ and $\mathfrak{sl}_n(\mathbb{C})$ are simple?
- Why are modular lattices important?
- Radii of convergence for complex series
- Proof involving Induction
- Can all rings with 1 be represented as a $n \times n$ matrix? where $n>1$.
- Nowhere monotonic continuous function
- Find a point position on a rotated rectangle