Tagged Questions

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$\vec{a} \times \vec{b} = \vec{c} \times \vec{d}$ . what can you say about the direction of $\vec{b} \times \vec{c}$?

I know that $\vec{a} \times \vec{b}$ and $\vec{c} \times \vec{d}$ are perpendicular therefore the dot product would equal $0$.
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Find parametric equations using parallel lines and line through a point

How would I find the parametric equation of a line through $(1,-1,1)$ and parallel to the line $x + 2 = 1/2y = z -3$. Would I find the vector equation first? If so, how would I go about doing that?
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Prove that if $f$ is linear, then for any $\textbf{a},\textbf{v} \in \mathbb{R}^2$, $f(\textbf{a}+\textbf{v})=f(\textbf{a})+[Df(\textbf{a})]\vec{v}$

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a function. Prove that if $f$ is linear, then for any $\textbf{a},\textbf{v} \in \mathbb{R}^2$, ...
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Find a vector parallel to the intersection of the planes $2x-3y+5z=2$ and $4x+y-3z=7$

The solution is $(4,26,14)$. I know how to find the intersection of the planes, but not a parallel vector.
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Does the map $F(x, y)=(f(x, y), y)$ induce an isomorphism $dF_{(a, b)}$?

Suppose $f:\mathbb R^{n+m}\longrightarrow \mathbb R^n$ be a $C^p$ map such that $df_{(a, b)}:\mathbb R^{n+m}\longrightarrow \mathbb R^n$ is surjective for some $(a, b)\in\mathbb R^{n+m}$. Define ...
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Is it true that $d\textbf{S} = dy dz\textbf{ i }+ dx dz\textbf{ j }+ dx dy\textbf{ k }$

I came up with this in my mind, Just wondering if it is true I am thinking about it too, will post my observations, if any
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Proof the change of variables theorem by volume comparison

My books prove the change of variables theorem by admitting a lemma (it says that linear algebra is needed so the proof won't be listed in the book): Let $\Psi:O\to \mathbb R$ be a smooth change of ...
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Any idea how to linearize this equation? $X^2-Y^2=aZ+bZ^2$

The intention is to linearize this equation $X^2-Y^2=aZ+bZ^2$ into something which looks like $Z=mX+nY+c$ so that a graph of $Z$ against $X$ or $Y$ can be plotted. X,Y,Z are variables while a,b,c are ...
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The derivative of matrix vector product with respect to matrix

Given function $$f(M) = Mv$$ where $M$ has dimension $n \times n$, and $v$ is a vector with dimension $n \times 1$. What's the derivative of $f(M)$ with respect to $M$?
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matrix differentiation - derivative of matrix vector dot product with respect to matrix

Given the function $$f(N) = x_1^T M x_2$$ where $x_1 = Nv_1$ $x_2 = Nv_2$ $x_1, x_2, v_1, v_2$ are vectors with dimension $n \times 1$ $M$ and $N$ are matrices with dimension $n \times n$ ...
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Acute angle between plane and line

Find the acute angle between: $x-y-3z=5$ and $x=2-t$ $y=2t$ $z=3t-1$ Here is how I proceed. I take the dot-product of the normal of the plane and the directional vector of the line. This gives me ...
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How do you solve a linear transformation with no transformation matrix given?

I am stuck, I can't see how Tff was found with no transformation matrix. And now am being asked to find Tgg, help me http://oi60.tinypic.com/33yrplv.jpg
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I need help with linear transforms? Linear Algebra [closed]

In the question below, how was [T]ff found? I have tried but I can't understand how because I usually start from a given matrix with variables, but non is given here. website is here; ...
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Why does this vector derivation hold?

I have the following variables/matrices: $$A \in \mathbb{R}^{m \times n} , \quad p \in \mathbb{R}^{n}, \quad \Sigma \in \mathbb{R}^{m \times m}, \quad w \in \mathbb{R}^{m}$$ where $\Sigma$ is a ...
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Do projections on $\mathbb{R}^2$ transform straight lines to straight lines?

A linear transformation $P:\mathbb{R}^{2} \longrightarrow \mathbb{R}^{2}$ is called projection if $P \circ P =P$. The question is: If $P$ is a projection then $P$ transforms straight lines ...
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Why generalize the derivative for multivariable functions? [duplicate]

Sorry if this is a dupe (did a search, couldn't find anything). In single variable calculus, if the following limit exists: $$\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h},$$ then this expression ...
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Solving a particular system of differential equations

The problem I'm trying to solve is this: $X'(t) \in \mathbb{R}^3 \,, \, \omega = (\omega_1,\omega_2,\omega_3)$ Find the general solution for $$X'(t) = \omega \times X(t)$$ After doing the cross ...
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Diagonal matrices and integrals

Suppose that $$A=\int_{\alpha}^{\beta} f(B,x)\ dx,$$ where $B$ is a $3\times3$ matrix. The result I'm looking for is that if $B$ is diagonalized with an orthogonal matrix, then is A diagonalized by ...
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Where do these Paths intersect

We have two paths: $r(t)=\langle cos(t),0,sin(t)\rangle$ $s(t)= \langle 0,cos(t),sin(t)\rangle$ where $t$ in $[0,pi]$ We are given that they are on the surface of $F(x,y,z)= x^2 + y^2 + z^2 -1$ ...
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Find the differential of $f(A)=det(A^{-1}-A)$ where $A$ is invertible.

The question is if $A$ is an invertible matrix with real entries of size $n$. Is $f(A)=det(A^{-1}-A)$ differentiable? and what is the differential. I think I managed to show it's differentiable. the ...
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Hessian of a square root of a quadratic form

What is the Hessian matrix of the square root of a quadratic form: $\left(w^T H w\right)^{0.5}$? Got the gradient, $0.5 \left(w^T H w\right)^{-0.5} ( 2 H w)$, which gives numerically correct ...
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Kalman's controllability condition implies that the r.h.s. is “non-degenerate.”

Assume $f:\mathbb R^n\times\mathbb R\to\mathbb R^n$ is given by $f(x,u)=Ax+bu$ for some choice of coordinates $(x,u):x\in\mathbb R^n,u\in\mathbb R$, some constant matrix $A\in M_n(\mathbb R)$, and a ...
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Limits of norms and deriviative as linear transformation

I'm self-studying Spivak's Calculus on Manifolds and he introduces the derivative by first looking at it as a linear transformation, $Df(a) = \lambda$, saying that for a differentiable function ...
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Determine Cross Product with Left Hand vs Right Hand

If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let: the 1st vector in the cross product = your index finger = in red ...
132 views

Product of Elements in SU(2)

Let $$V := \frac{x_4+i\vec{x}\cdot{\vec{\sigma}}}{\left|x\right|}$$ where $\left(x_1,x_2,x_3,x_4\right)\in\mathbb{R}^4$, $|x|$ is the Euclidean norm, and $\sigma^j$ are the Pauli matrices. Let ...
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How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
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Solution in common for two differential equations

Consider: $E1: y''-4y'+4y=0$ Solution: $y(x)=c_1 e^{2x}+c_2 x e^{2x}$ $E2: y''-2ay'+(a^2-1)y=0$ Solution: $y(x)=c_1 e^{(a+1)x}+c_2 e^{(a-1)x}$ For what values of $a$, $E1$ and $E2$ have ...
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How to find an equation of the plane, given its normal vector and a point on the plane? [duplicate]

I have a question regarding vectors: Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $A(1,5,3)$. (A cartesian and parametric ...
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Powers of traces, integrals over spheres and class functions

Let $V$ be a complex vector space of dimension $\operatorname{dim}_{\mathbb C} V = n$, equipped with a Hermitian inner product $\langle \,\cdot\,,\,\cdot\, \rangle$. Let also $A$ be an endomorphism of ...
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A Question about Non-Conservative Vector Fields

In my multivariable calculus class, we spent some time discussing the vector field that was the gradient of arctan(y/x). This field was shown to be non-conservative in closed regions which enclosed ...
Use the definition of a negative definite matrix to show that if A is negative semi-definite: $$A_{ii} ≤ 0 \ \forall i$$ I know the definition (in terms of quadratic form) and the equivalent rules ...