# Tagged Questions

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### explain this confusing algebraic identity?

Can anyone show, step-by-step, how the expression on the LHS can be turned into the expression on the RHS? $x^ay^b=a^ab^b(a+b)^{-(a+b)}(x+y)^{a+b}$
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### Going from Linear algebra to Multivariable Calculus [closed]

I just finished a course in Linear algebra, can anyone tell me how Linear and multivariable calc are related?
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### Acquiring $Df(\mathbf{x})$

Sorry for the probably easy and silly question, but I try to teach myself linear algebra and I am stucked at "the derivative as a matrix" part. I know how to differentiate partially and I know how ...
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### Why is this proof valid - inverse function theorem

Question from worksheet, I don't fully understand the solution the teacher gave. Question: let $S$ be the set of symmetric positive definite matrices of dimension $n$x$n$. Let $T: S \to S$, ...
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### Proving $\nabla_A tr(ABA^T C) = CAB + C^T A B^T$

The above equation appears without proof on page 9 (equation 3) of Andrew Ng's notes on Machine Learning I have tried various approaches to prove this to no avail. From the notes it seems that it ...
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### When is a function of two variables positive?

There are two functions, $u(t)$ and $v(t)$, $u:[0,\infty]\to {R}$ , $t=\frac{1}{2}(y_{1}^{2}+y_{2}^{2})$, $v=\frac{-uu'}{2tu'-u}$. They should satisfy the next: $u>0$ and $u+2tv>0$. If we ...
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### How do you rearrange equations with dot products in them?

How can I go about rearranging an equation similar to this... $$\left(\pmatrix{-3\\0\\1}+ t \pmatrix{1\\4\\7} \right) \cdot n - a = 0$$ The issue I'm having is manipulating dot products
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### multiplying Gaussian distributions of different dimensions

The multiplication of multivariate Gaussian distributions defined over some parameter vector of a given dimension can be achieved by the following. Assuming that the Gaussian is parametrized by the ...
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### How to compute the Jacobian Matrix of the next system?

I have a little problem with notation and I do not know how to work it out. I have the next system $$\dot x = A(x)x,$$ where $x \in \mathbb{R}^n$ and the square matrix $A(x)_{n\times n}$ has ...
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### Volume of ellipsoid using Linear Algebra

Can someone tell me how to find the volume of an ellipsoid of dimension $\mathbb{R}^3$ by using linear algebra? I know the formula is $\frac{4}{3}\pi abc$. I am given the equation ...
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### Weird result with Multivariate Normal Distribution

Take zero mean MVN on k dimensions: $$p(x) = \frac 1 {\sqrt {( 2 \pi ) ^k |\Sigma|}}e^{-\frac {1} {2} x \Sigma^{-1} x}$$ we will surely agree that $p(x)\leq1$ for all $x$, including $x=0$ ...
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### Why is this 'obviously' positive semi-definite?

Here is a snapshot from a book I am studying. I learned all about positive semi-definiteness, and in fact I know that this matrix they are showing is in fact PSD. What I do not know is how they ...
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### Expressing the determinant of a sum of two matrices?

Can $$\det(A + B)$$ be expressed in terms of $$\det(A), \det(B), n$$ where $A,B$ are $n\times n$ matrices? # I made the edit to allow $n$ to be factored in.
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### Directional derivatives, linear maps, and uniform convergence

The Exercise Let $f(x,y)=x$ if $|y|>x^2$ and $f(x,y)=0$ otherwise. Show that all the directional derivatives of $f$ exist at the origin but there does not exist a linear map $D$ such that ...
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### Proofs using vector properties

Let $a,b$ and $c$ be vectors in $\mathbb{R}^3$. How do I show that $$\|a-b\| \le \|a-c\|+\|c-b\|$$ and $$\|a \times b\|^2=\|a\|^2\|b\|^2-(a\cdot b)^2$$ ?
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### the space of exterior k-forms is infinite dimensional. why?

let Z be an n-dimensional smooth manifold with smooth (n−1)-dimensional boundary ∂Z, representing the space of spatial variables. Denote by $Ω^k$(Z), k = 0, 1, . . ., n, the space of exterior k-forms ...
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### Derivative of a Linear Map

I'm devastatingly incompetent at linear algebra and multivariable calculus. I just cannot understand it at all. Here's the easiest problem from my homework, and my attempt at solving it, and where I ...
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### Partial Differentiation Question. Solving when there is many variables

In one of my computer science classes we were given a homework problem that deals with partial differentiation. I never learned this in my math classes and have been trying to teach myself this but ...
I'm asked to find if there is any point of intersection, and if so, where it is between the line represented by the symmetric equation $\frac{x-3}{3}=\frac{y+1}{-2}=\frac{z-10}{4}$ and the plane ...