Main thing of interest for me is that the determinant in that case is $$\det M = 1+2\cos \theta_{ab}\cos \theta_{bc}\cos \theta_{ac}-\cos^2 \theta_{ab}-\cos^2 \theta_{bc}-\cos^2 \theta_{ac}$$
No, I actually still have unit vectors in the present problem. But now instead of having {a,b,c} and their pairwise dot products, I have {a,b}, {c,d} and the dot products between these sets
Given a regular pentachoron ($ABCDE$), find the cross section when it is cut along a hyperplane which is equidistant from and parallel to line $AB$ and plane $CDE$.
Here, a regular pentachoron means a $5-$ cell, i.e., a $4D$ figure with $5$ vertices A,B,C,D,E all at equal distances from o...
Given a regular pentachoron ($ABCDE$), find the cross section when it is cut along a hyperplane which is equidistant from and parallel to line $AB$ and plane $CDE$.
Here, a regular pentachoron means a $5-$ cell, i.e., a $4D$ figure with $5$ vertices A,B,C,D,E all at equal distances from o...
@TedShifrin because according to wiki, $\mathrm d(f \ \mathrm dt) = (\mathrm df)(\mathrm dt)$, implying that the second term generated by the product rule ($f \ \mathrm d^2t$) vanishes
@Tanuj: You should figure out that there's no problem with the second derivative of that function at $0$. You have to do similar things with the more complicated part. You have to find $f'(x)$, $f'(0)$, and use the definition of the derivative.
let's say I know that for every proper ideal $P$ there's a maximal ideal $M \supseteq P$, and that for every ideal $I$ contained in a prime ideal $Q$, there is a prime ideal $J$ that is minimal with respect to the property $I \subseteq J \subseteq Q$. I wonder if I can prove the following without Zorn's lemma: for every element $f$ of the ring and ideal $I$, if $f^n \notin I$ for all $n$, then there is a prime ideal containing $I$ not containing $f$.
@AkivaWeinberger, sorry for asking this very late but, why is no subsequential limit of the partial sums of (25) has limit less than $\alpha$ or greater than $\beta$ here?
I tried doing this way, but it seemed that then every subsequential limit of the partial sums of (25) would be $\alpha$ or $\beta$ and nothing between them, which seemed unnatural.
@robjohn, Why is this on hold? What part of the question is unclear? Will it get deleted? What part of the question is unclear?
Let $F\subset E$ a vector subspace, such that $\overline{F}\neq E.$ There exists a functional $f\in E^*$ such that $$\langle f,x\rangle=0, ~ x\in F,$$ but $f\not\equiv 0.$
@Vrouvrou take $v \in E \setminus \overline{F}$, define a functional on $v \oplus \overline{E}$ that takes $v$ to $1$ and is $0$ on $\overline{E}$, then extend by Hahn-Banach
Hey there! I'm taking a Linear Algebra practice test and am stuck on a question. I wanted to know what I'm doing wrong without looking at the exact solution. The question is Let A = [column1(5, -1), column2(-1, 5)]. Compute a formula for A^k where k is a positive integer. Your answer should be a single matrix.
So I did A^2 and got [col1(26, -10), col2(-10,26)], but figured repeated multiplication was not the way to go, so I used the eigenvalues and eigenvectors to diagonalize A to [col1(4, 0), col2(0, 6)] with the change of basis matrix [col1(1, 1), col2(-1, 1)]. Then I inverted the change of basis matrix to [col1(1/2, -1/2), col2(1/2, 1/2)].
I multiplied [col1(4^k, 0), col2(0, 6^k)] on the left by this inverse to get the putitive solution [col1((4^k)/2, -(4^k)/2), col2((6^k)/2, (6^k)/2)], but it does not satisfy the above result I got when initially computing A^2. Am I on the right track with a computation error, or am I using an invalid approach?
It's the statement if $Y \subset X$ is a closed subspace and $x_0 \notin Y$, then there is a $x' \in X'$ with $x'=0$ on $Y$ and $\|x'\|=1$ and $x'(x_0)=\operatorname{dist}(x_0,Y)$
Given a regular pentachoron ($ABCDE$), find the cross section when it is cut along a hyperplane which is equidistant from and parallel to line $AB$ and plane $CDE$.
Here, a regular pentachoron means a $5-$ cell, i.e., a $4D$ figure with $5$ vertices A,B,C,D,E all at equal distances from o...
