Linear Definition and 1000 Threads

Standard matrix for T is: $$P=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & -1 \end{bmatrix}$$ (i) Since matrix P is already in reduced row echelon form and each row has a pivot point, ##T## is onto mapping of ##\mathbb R^3 \rightarrow \mathbb R^2## (ii) Since there is free variable in matrix P, T is...

We have a slide in class that states if no friction or damping force, then the system is conservative. Then it shows: delta(T+U)=0 or T+U=constant. It then goes on to say that max kinetic energy is equal to max potential energy which is false. no way can you have KEmax=Pemax... I double...

Attempt of a solution. By the Rank–nullity theorem, $$ \dim V=\dim Im_{F}+\dim\ker\left(F\right) \Rightarrow n=1+\dim\ker\left(F\right) \Rightarrow \dim\ker\left(F\right)=n-1. $$ Similarly, it follows that $$\dim\ker\left(G\right)=n-1.$$ This first part, for obvious reasons, is very clear. The...

A space is infinite dimensional when its basis is infinite. But how can I ensure that the basis of the space of all sequences whose limit is zero is infinite? (After solving that, I would like to have a hint on this very similar problem which says: let V be a Linear space of all continuous...

I'm looking at the following web page which looks at rendering bezier curves. GPU Gems 3 - Chapter 25 Paper on same topic The mathematics is quite interesting, I was interested to know what the F matrix would look like for for a linear bezier equation. The maths for the quadratic case is in...

Hi Everybody, I am having some difficulties on the prove this problem. I picked a nice example when I was trying to think about the proof. Let ##s=3## and ##t=2##. Then ##u1=c1v1+c2v2, u2=c3v1+c4v2, u3=c5v1+c6v2##. Then a linear combination of u: ##K1u1+K2u2+K3u3=0##. I grouped both linear...

We got two vectors ##\mathbf{v_1}## and ##\mathbf{v_2}##, their sum is, geometrically, : Now, let us rotate the triangle by angle ##\phi## (is this type of things allowed in mathematics?) OC got rotated by angle ##\phi##, therefore ##OC' = T ( \mathbf{v_1} + \mathbf{v_2})##, and similarly...

We have a transformation ##T : V_2 \to V_2## such that: $$ T (x,y)= (x,x) $$ Prove that the transformation is linear and find its range. We can prove that the transformation is Linear quite easily. But the range ##T(V_2)## is the the line ##y=x## in a two dimensional (geometrically) space...

Is Advanced Linear and Matrix Algebra by Nathaniel Johnston a good book on linear algebra? Will it teach me all I need to know? Is there any calculus in it despite the name? I never took a course on linear algebra so I'm looking for something that teaches everything and includes calculus with...

I am trying to understand the effect of relativistic length contraction on the electron bunches in a linear accelerator. Figure B is for nonrelativistic speeds, successive cylinder lengths are progressively longer. However, wikipedia says "At speeds near the speed of light, the incremental...

Suppose we have V, a finite-dimensional complex vector space with a Hermitian inner product. Let T: V to V be an arbitrary linear operator, and T^* be its adjoint. I wish to prove that T is diagonalizable iff for every eigenvector v of T, there is an eigenvector u of T^* such that <u, v> is...

I have a sequence of functions ##0\leq f_1\leq f_2\leq ... \leq f_n \leq ...##, each one defined in ##\mathbb{R}^n## with values in ##\mathbb{R}##. I have also that ##f_n\uparrow f##. Every ##f_i## is the limit (almost everywhere) of "step" functions, that is a linear combination of rectangles...

Or is reading a proofs book enough

Now i learned how to use discriminant i.e ##b^2-4ac## and in using this we have; ##b^2-4ac##=##0-(4×3×2)##=##-24<0,## therefore elliptic. The textbook has a slight different approach, which i am not familiar with as i was trained to use the discriminant at my undergraduate studies... see...

Problem: Given the line L: x = (-3, 1) + t(1,-2) find all x on L that lie 2 units from (-3, 1). I know the answer is (3 ± 2 / √5, -1 ± 4/√5) but I don't know where to start. I found that if t=2, x= (-5, 5) and the normal vector is (2, 1) but I am not sure if this information is useful or how...

Hi all, I'm opening this thread because of my uncertainty in how to correctly approach this exercise. My first thought was that, since the plate is subject to friction with the floor, it is going to stop, thus the final moment is 0. Hence, from the conservation of linear moment: $$m_Av_A+\sum...

Given ## a,b,c,d,e,f \in \mathbb {R}, ad - bc \neq 0 ##, if ##(x_1,y_1)## and ##(x_2,y_2)## are pairs of real numbers satisfying: ## ax_1 + by_1 = e, cx_1 + dy_1 =f ## ## ax_2 + by_2 = e, cx_2 + dy_2 = f ## then ## (x_1,y_1) = (x_2,y_2). ## Here is my attempt at a proof, I have gotten stuck...

All polymer linear bearings i have seen have a lot of groves in the running direction (at least the ones for round shafts). Is this just to minimize the surface area in contact with the shaft to minimize the friction? Or do they have another function like clearing dust and particles? And why...

Let ##m_s = 0.05, m_{s_1} = 0.02, m_r = 0.12, L = 0.8.## be the masses of the two spheres, mass of the rod, and length of the rod. Then the work done by gravity when the rod reaches the vertical position is ##(m_s(L/2) - m_{s_2}(L/2))g## and the kinetic energy equals ##\frac{1}2 (\frac{1}{12}...

ANSWER: element x does not present a linear resistance because it isn't constant as i and v increases. Is my answer correct?

First of all, I attached pictures of the very last algebra textbook that I have finished studying. I'm going the self taught route. I really loved this book because it had lots of examples, practice exercises, quizzes and even tests! It also had answers in the back. It's currently my favorite...

I have a given point (vector) P in R^3 and a 2-dimensional linear subspace S (a plane) which consists of all elements of R^3 orthogonal to P. The point P itself is element of S. So I can write P' ( x - P ) = 0 to characterize all such points x in R^3 orthogonal to P. P' means the transpose...

The correct answer is: #P = \int \frac{dp^3}{(2\pi)^3}\frac{1}{2E_{\vec{p}} \big(a a^{\dagger} + a^{\dagger}a\big)# But I get terms which are proportional to ##aa## and ##a^{\dagger}a^{\dagger}## I hereunder display the procedure I followed: First: ##\phi = \int...

Hello, I have used an edge current of 10 A through a 0,45 cm (lenght) wire inside an air sphere. The thing is that, according with Ampere law, the magnetic field (B) produced at a 1 mm of distance from the wire shall be 0,002 T, and I am obtaining much higher values in this simulation (around...

This is another open ended question, exploring a space of design concepts, in similar spirit to this. I want to explore monopods with regard to travel in densely populated cities(even possibly intercity travel). The main idea is to use small personalized pods to travel in tubes(or tracks). The...

I'm watching this minutephysics video on Lorentz transformations (part starting from 2:13 and ending at 4:10). In my spacetime diagram, my worldline will be along the ##ct## axis and the worldline of an observer moving relative to me will be at some angle w.r.t. the ##y## axis. When we switch...

Let ##G\leq GL(n)## be a linear algebraic group of dimension ##m,## and ##C## its ##c##-dimensional center. What do we know about lower and upper bounds of ##c=c(m)\,\text{?}## Clearly ##c(0)=0, c(1)=1## and ##n^2\geq c(m)\geq 1## for ##m\neq 0.## By Schur's Lemma we also know ##c(n^2)=1##. Did...

https://www.testsite.cocoams.org/linrec.pdf

I have attached my work to this thread. Could someone help me with this Linear Algebra problem. This is my first week so I do not know many advanced ways to solve these problems. I could not figure out how to get this matrix into rref, so I solved it the following way. Is the way I used...

How does a 3 stage linear actuator mechanical work. I can only find a regular linear actuator mechanical but I'm unsure how will the last stage go up and down. Anyone got a poor 3d drawing for a better understanding.?

I am currently trying to create a linear induction motor for fun and am having some trouble getting it to start oscillating or move at all. I am using this video as a reference... I am using 3D printed PLA as the structure for the copper to wind around, 26 GA Craftware USA copper wire, 5/8"...

Consider the second order linear ODE with parameters ##a, b##: $$ xy'' + (b-x)y' - ay = 0 $$ By considering the series solution ##y=\sum c_mx^m##, I have obtained two solutions of the following form: $$ \begin{aligned} y_1 &= M(x, a, b) \\ y_2 &= x^{1-b}M(x, a-b+1, 2-b) \\ \end{aligned} $$...

First thing to notice is that ##L## and ##L \circ L## are precisely equal linear maps. What we know $$L \ \text{is injective} \iff \ker(L)=\{0\}$$ $$\ker L' = \{ x \in \Im(L) \ | \ L'(x)=0\}$$ $$\Im(L)=\{ x \in V \ | \ \exists \ v \in V \ \text{such that} \ L(v)=x\}$$ Besides, we notice...

I found this question in a textbook, not sure if this question has been asked before. Not sure if the author just wanted to make the reader think or he had anything specific in mind that he wanted the readers to understand. Most of the people immediately conclude that the speed of light doesn't...

Design a linear phase low pass filter whose specification is:1) Maximum tolerance in the passband equal to 0.01% (linear) in the passband;2) Cutoff frequency at ω_c = 0.3π and transition band at 0.05π;3) Minimum reduction of the 0.95 rejection band.

We only worry about finite vector spaces here. I have been taught that a subspace ##W## of a vector space ##V## has a complementary subspace ##U## if ##V = U \oplus W##. Besides, I understand that, given a finite vectorspace ##(\Bbb R, V, +)##, any subspace ##U## of ##V## has a complementary...

I am struggling to understand shocks in a one dimensional lattice with a linear spring connecting the masses. Say I have a one dimensional lattice with a linear spring constant, k and lattice spacing a. If the particles in the lattice has mass, m then my speed of sound c is a*sqrt(k/m). That is...

Show that ##U = span \{ (1, 2, 3), (-1, 2, 9)\}## and ##W = \{ (x, y, z) \in \Bbb R^3 | z-3y +3x = 0\}## are equal. I have the following strategy in mind: determine the dimension of subspaces ##U## and ##W## separately and then make use of the fact ##dim U = dim W \iff U=W##. For ##U## I would...

Hello all, I have a problem related to LU Factorization with my work following it. Would anyone be willing to provide feedback on if my work is a correct approach/answer and help if it needs more work? Thanks in advance. Problem: Work:

Hi guys! :) I was solving some linear algebra true/false (i.e. prove the statement or provide a counterexample) questions and got stuck in the following a) There is no ##A \in \Bbb R^{3 \times 3}## such that ##A^2 = -\Bbb I_3## (typo corrected) I think this one is true, as there is no squared...

I have a machine I am designing that for all intensive descriptions, is a simple press designed to compress loose product into a puck like shape. The press force comes from a roller bearing mounted to a piston shaft, the rod sliding through a rigid linear bearing and the piston on the end of...

A rumour spreads through a university with a population 1000 students at a rate proportional to the product of those who have heard the rumour and those who have not.If 5 student leaders initiated the rumours and 10 students are aware of the rumour after one day:- i)How many students will be...

hi guys I was trying to find the matrix of the following linear transformation with respect to the standard basis, which is defined as ##\phi\;M_{2}(R) \;to\;M_{2}(R)\;; \phi(A)=\mu_{2*2}*A_{2*2}## , where ##\mu = (1 -1;-2 2)## and i found the matrix that corresponds to this linear...

The average weight of a male child’s brain is 970 grams at age 1 and 1270 grams at age 3. (Source: American Neurological Association) (a) Assuming that the relationship between brain weight y and age t is linear, write a linear model for the data. (b) What is the slope and what does it tell...

Write a linear equation. A school district purchases a high-volume printer, copier, and scanner for $24,000. After 10 years, the equipment will have to be replaced. Its value at that time is expected to be $2000. Write a linear equation giving the value V of the equipment during the 10 years it...

Write a linear equation for the application. A pharmaceutical salesperson receives a monthly salary of $5000 plus a commission of 7% of sales. Write a linear equation for the salesperson’s monthly wage W in terms of monthly sales S. Solution: I am looking for W(S). S = monthly sales Let...

You are given the dollar value of a product in 2016 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value V of the product in terms of the year t. (Let t = 16 represent 2016.)1...

The average weight of a male child’s brain is 970 grams at age 1 and 1270 grams at age 3. (Source: American Neurological Association) Assuming that the relationship between brain weight y and age t is linear, write a linear model for the data. I am going to start by saying y = mx + b. Let x...

A school district purchases a high-volume printer, copier, and scanner for $24,000. After 10 years, the equipment will have to be replaced. Its value at that time is expected to be $2000. Write a linear equation giving the value V of the equipment during the 10 years it will be in use. Let t =...