2 votes
Accepted

Geometry problem inspired by Babylonian tablets (positive matrix factorization)

quite a mess. Given positive definite quadratic form $a x^2 \pm 2bxy + c y^2 $ so that $ac>b^2,$ take $r > 0,$ with $$ 0 < r < \min \left( \frac{b}{a}, \frac{c}{b} \right) $$ Then ...
user avatar
  • 130k
2 votes
Accepted

Why am I getting t is a vector when solving for t in the vector equation of a 3d line.

The trouble comes when you go from $$(x-x_0, y-y_0, z-z_0) = t(a,b,c)$$ to $$t = \frac{(x-x_0,y-y_0,z-z_0)}{(a,b,c)} = \big(\frac{x-x_0}{a}, \frac{y-y_0}{b}, \frac{z-z_0}{c}\big).$$ We cannot divide ...
user avatar
2 votes

Finding how to decompose a polynomial with respect to a polynomials' basis

$x^2+x^3,\; \mathbf{B} = \langle \,\underbrace{1}_{p_1}\,, \,\underbrace{1+x}_{p_2}\,, \,\underbrace{1+x+x^2}_{p_3}\,, \,\underbrace{1+x+x^2+x^3}_{p4}\, \rangle$ This one is "easily solved by ...
user avatar
  • 68.6k
2 votes
Accepted

Determining a value to form a projector

In your second-to-last step, you got rid of the left side but did nothing to the right side, which is a mistake. After that, it needs to be clearer why you're concluding $2x-1=0$ from your projector ...
user avatar
  • 14.8k
1 vote

Blend multiple shapes.

Finally got it working. pseudocode: ...
user avatar
1 vote

Geometry problem inspired by Babylonian tablets (positive matrix factorization)

From a different angle, let $\,u=x + iy\,$, $\,v=w + iz\,$, then the equations are $\,|u|^2=a\,$, $\,|v|^2 = c\,$, $\,\text{Im}(uv) = b\,$. The latter can be written as $\,u v - \bar u \bar v = 2ib\,$,...
user avatar
  • 68.6k
1 vote
Accepted

Let f be a correspondence between vector spaces V& W. Show that the spaces V & W are isomorphic via f if and only if (rest of question inside) ...

Let $B=\{\vec{b}_1,\vec{b}_2,\ldots, \vec{b}_n\}$ and $D=\{\vec{d}_1,\vec{d}_2,\ldots, \vec{d}_n\}.$ According to the assumptions for any $\vec{v}$ we have $$\vec{v} =\sum_{k=1}^n \alpha_k\vec{b}_k,\...
user avatar
1 vote

Function That Weights Toward End of Range [0,1]

Let's consider functions of the form $f(x) = a(x - 0.5)^b + 0.5$, with the constraints that $f(0) = 0$ and $f(1) = 1$. Then: $$f(0) = a(-0.5)^b + 0.5 = 0 \implies a(-0.5)^b = -0.5$$ $$f(1) = a(0.5)^b ...
user avatar
  • 7,704
1 vote
Accepted

I want to find $m,l,n$ and $k$ and $(A_1,B_1),(A_2,B_2)\in V\times W$ such that $A_1B_1+A_2B_2\neq A_3B_3$ for any $(A_3,B_3)\in V\times W$.

Another solution that also uses spaces of matrices, but which is more simple and valid for any field $k$. It is based on the equality: $$\begin{pmatrix} 1\\ 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \...
user avatar
1 vote

Invertibility of linear maps (Exercise 3D.4 of Axler's "Linear Algebra Done Right")

Call $K$ the kernel (aka null space) of $T_1$ and $T_2$. Both ranges $T_1V$ and $T_2V$ are isomorphic to $V/K$ by the first-isomorphism theorem (the rank-nullity theorem is just the dimensional ...
user avatar
  • 14.8k
1 vote
Accepted

Compact SVD for $A_{i, j} = \sin(i) \cdot \cos(j)$

They are already given to you. Just take $$a=\sqrt{\sum_{i=0}^{n-1}\sin^2i},\qquad\qquad b=\sqrt{\sum_{j=0}^{n-1}\cos^2j},$$ $$ U=\frac1a\begin{bmatrix}\sin 0\\ \vdots\\ \sin(n -1)\end{bmatrix},\qquad\...
user avatar

Only top scored, non community-wiki answers of a minimum length are eligible