0
votes
2answers
52 views

$\vec{a} \times \vec{b} = \vec{c} \times \vec{d}$ . 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 $0$.
-1
votes
1answer
24 views

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?
1
vote
1answer
16 views

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 $\textbf{a},\textbf{v} \in \mathbb{R}^2$, ...
0
votes
2answers
48 views

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.
0
votes
1answer
16 views

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 ...
1
vote
1answer
59 views

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
0
votes
1answer
35 views

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 ...
2
votes
2answers
93 views

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 ...
0
votes
0answers
29 views

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 $M$?
0
votes
1answer
43 views

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 $n \times n$ ...
1
vote
0answers
24 views

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 ...
0
votes
0answers
42 views

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
0
votes
1answer
43 views

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; ...
2
votes
2answers
68 views

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 ...
1
vote
2answers
50 views

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 ...
4
votes
1answer
79 views

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 ...
3
votes
2answers
95 views

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 ...
1
vote
1answer
30 views

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 ...
1
vote
0answers
58 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
0
votes
1answer
39 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
1
vote
1answer
41 views

How to factor and reduce a huge determinant to simpler form? Linear Algebra

So, I have learned about cofactor expansion. But the cofactor expansion I know doesn't reduce the number of rows and colums to one matrix. I usually pick a colum, multiply each element in the column ...
0
votes
3answers
26 views

How do you know that rows are independent and what are the 120 terms?

I am having trouble with the question below, help me out;
0
votes
2answers
56 views

Suppose $(x,y,z)$, $(1,1,0)$, and $(1,2,1)$ lie on a plane through the origin.

What determinant is zero? What equation does this give for the plane? I need some help here, am pretty stuck
1
vote
1answer
30 views

Derivation of Mahalanobis Distance

I was recently reading up on the Mahalanobis Distance, and understood how it generalizes distance measures for multivariate data such as the Euclidean Distance. However, what got me wondering was how ...
0
votes
2answers
62 views

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? $x^ay^b=a^ab^b(a+b)^{-(a+b)}(x+y)^{a+b}$
2
votes
1answer
25 views

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 ...
0
votes
1answer
31 views

Derivative of a path.

Define path: $s(t)= \langle 0,\cos(t),\sin(t)\rangle$ We are given that it is on the surface of $F(x,y,z)= x^2 + y^2 + z^2 -1$ Am told to find $s'(t)$ and a given point $t = (\pi/2)$. Is $s'(t) = ...
0
votes
1answer
17 views

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 $F(x,y,z)= x^2 + y^2 + z^2 -1$ ...
3
votes
0answers
17 views

$\sqrt{X}$ where $X$ is a positive definite matrix is smooth $C^{\infty}$ [duplicate]

I'm trying to prove the following statement. Let $P_n \subset Mat_{nxn}(\mathbb R)$ be the set of all symmetric positive definite matrices with real entries of size $n$x$n$. Let $\sqrt{}:P_n \to ...
0
votes
1answer
39 views

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 $T: S \to S$, ...
2
votes
0answers
21 views

Rank of the differential

Let $f:\mathbb R^n \to \mathbb R^n$ such that $f$ maps roots of a polynomial to its coefficients. Meaning: if $(x-x_1)(x-x_2)...(x-x_n)=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_n$ then $f\begin{pmatrix} x_1 ...
2
votes
1answer
42 views

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 ...
0
votes
1answer
43 views

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 ...
1
vote
1answer
29 views

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 ...
0
votes
0answers
24 views

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 ...
2
votes
1answer
36 views

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 ...
4
votes
1answer
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 ...
0
votes
1answer
79 views

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 ...
2
votes
1answer
35 views

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 ...
1
vote
1answer
30 views

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 ...
0
votes
0answers
28 views

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 ...
2
votes
1answer
65 views

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 ...
0
votes
1answer
28 views

Elements of a Negative Semidefinite Matrix

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 ...
3
votes
1answer
46 views

Cross Product in Levi-Civita Notation - The elementary basis vector's missing?

http://www.unl.edu.ar/ceneha/uploads/Cartesian_tensors_Index_notation_&_summation_convention.pdf avers: $1.$ $(a×b).(c×d) = \epsilon_{i jk}a_jb_k \quad e_{ilm}c_ld_m$ $2. \nabla × ...
1
vote
2answers
52 views

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 ...
0
votes
0answers
19 views

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 ...
4
votes
2answers
164 views

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
0
votes
0answers
29 views

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 ...
0
votes
0answers
38 views

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 ...
2
votes
1answer
99 views

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 ...