3 votes
Accepted

If $B = x(xI-A)^{-1}$ for a generator matrix $A$, then $B-B^2$ has positive diagonal elements

Let me continue from @user1551's idea and show that indeed $\mathbf{B} - \mathbf{B}^2$ has non-negative diagonals. Step 1. We recap @user1551's reduction. Let $\mathbf{A}$ be a transition-rate matrix ...
Sangchul Lee's user avatar
3 votes
Accepted

Defining a Plane with Two Parallel Lines

Let the first line be $a_0 + tv$ and let the second line be $a_1 + tv$ where $t$ is a parameters. Then your plane contain the third line $a_0+s(a_1-a_0)$ where $s$ is a parameter. Now you have found ...
Siong Thye Goh's user avatar
2 votes

Defining a Plane with Two Parallel Lines

A plane can be defined like this $$ E: \vec{x}=\vec{p}+r\vec{a_1}+t\vec{a_2}. $$ Here, every point $x$ can be reached via a vector $\vec{p}$ that goes from the origin to any point $P$ on the plane. ...
calculatormathematical's user avatar
2 votes

Find a vertex of a tetragon where three vertices are given

Hint. Calling $\hat{k}=\frac{(V-W)\times(W-U)}{\|(V-W)\times(W-U)\|}$ we can determine $\hat p,\hat q$ such that $$ \cases{ \hat p\cdot \hat k = 0\\ \|\hat p\| = 1\\ \hat q = \hat k\times \hat p } $$ ...
Cesareo's user avatar
  • 32.6k
2 votes

Skew Symmetric Matrix for expressing a Rotation

Rotations in $\Bbb{R}^3$ can be represented by the orthogonal matrices of the group $SO(3)$. Since it is a Lie group, the elements of the associated Lie algebra $\mathfrak{so}(3)$ play the role of ...
Abezhiko's user avatar
  • 7,250
2 votes

Show: $\text{im } f \cap \text{ker } f = \{0\} \iff \text{ker } f \circ f = \text{ker } f$ for vectorspace $V$ with linear function $f:V\rightarrow V$

I think it will be clearer to write separately. $\Rightarrow$ Obviously,we have $\ker f\subseteq \ker f\circ f$.So we just need to prove $\ker f\supseteq \ker f\circ f$.For every $x\in \ker f\circ f$,...
RainField's user avatar
  • 149
1 vote

About vector spaces over finite fields

Let $\bar{x}\in\Bbb{F}_q^n$ be a vector in a finite-dimensional vector space over a finite field. Let $x_i\in\Bbb{F}_q$ be a 'meaningful' coordinate. If $x_i\neq0$ then we can scale the vector so that ...
Servaes's user avatar
  • 62.8k
1 vote

Symmetric group representation $S_{3}$ to $\mathbb{C}^{3}$

Your mistake is in the equality $$ (x_{\mu^{-1}\circ \sigma^{-1}(1)},x_{\mu^{-1}\circ \sigma^{-1}(2)},x_{\mu^{-1}\circ \sigma^{-1}(3)}) = g_\mu(x_{\sigma^{-1}(1)},x_{\sigma^{-1}(2)},x_{\sigma^{-1}(3)})...
Albert's user avatar
  • 2,732
1 vote
Accepted

Proving $f(V)=\mathrm{span}(f(\mathbf{v}_{1}),\dots, f(\mathbf{v}_{n})).$

Julio Puerta basically answers your precise question in the comment section. Yet, you could lighten your proof and make it more readable (and also circumvent your question) by considering a single ...
Abezhiko's user avatar
  • 7,250
1 vote

How to prove the following determinant identity

We will execute a "degree matching argument". That is, The determinant of the matrix $$ \begin{array}{|cccccccccc|} 1 & 0 & 0 & \cdots & 0 & 1 & 0 & 0 & \...
Sarvesh Ravichandran Iyer's user avatar

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