Linear Definition and 1000 Threads

Problem: Let $T$ be the linear operator on $\mathbb{R}^3$ defined by $$T(x_1, x_2, x_3)= (3x_1, x_1-x_2, 2x_1+x_2+x_3)$$ Is $T$ invertible? If so, find a rule for $T^{-1}$ like the one which defines $T$. Prove that $(T^2-I)(T-3I) = 0.$ Attempt: $(T|I)=\left[\begin{array}{ccc|ccc} 3 &...

For the brief explanation: $\mathcal{P}$ contains $0$ by choice $p(x) = 0$ and polynomial plus a polynomial is a polynomial, and a scalar times a polynomial is a polynomial. So $\mathcal{P}$ is a non-empty subset of $\mathcal{C}^{\infty}$ that's closed under addition and scalar multiplication...

1. Show that the map $\mathcal{A}$ from $\mathbb{R}^3$ to $\mathbb{R}^3$ defined by $\mathcal{A}(x,y,z) = (x+y, x-y, z)$ is a linear transformation. Find its matrix in standard basis. 2. Find the dimensions of $\text{Im}(\mathcal{A})$ and $\text{Ker}(\mathcal{A})$, and find their basis for the...

Here is a linear version of the Ehrenfest paradox with the goal of understanding the observations of someone in motion in the scenario, then solicit your views on whether the calculations are correct and whether one can extend it to circular motion.Consider a one dimensional train of proper...

Homework Statement For the system of springs a) Assemble the stiffness matrix K and the force-displacement relations, K*u = f b) Find the L*D*L^T factorization of K. Use Matlab to solve c) Use the boundary conditions and applied forces to find the displacements Homework EquationsThe Attempt...

Homework Statement I am given the follow graph and asked to find the left null space Homework EquationsThe Attempt at a Solution First I start by transpose A because I know that the left null space is the null space of the incidence matrix transposed. I then reduce it to reduce row echelon...

Hi people, I studying electromagnetic waves (intermediate) and I don't understand how the expression for linear momentum of a wave is obtained, if the wave doesn't carry any mass. In particular, I have to explain why the radiation pressure on a perfect absorber is half that on a perfect...

I'm confused about the notation T:R^n \implies R^m specifically about m. From my understanding if n=2 then (x1, x2). Are we transforming n=2 to another value m for example (x1, x2, x3)?

I'm asked to check whether $\left\{1, e^{ax}, e^{bx}\right\}$ is linearly independent over $\mathbb{R}$ if $a \ne b$, and compute the dimension of the subspace spanned by it. Google said the easiest way to do this is something called the Wronskian. Is this how you do it? The matrix is: $...

I'm trying to understand the concept of vectors. Vectors have magnitude and a direction. When I read vector with some values \textbf{x} = \left(\begin{array}{c}x_1\\x_2\\x_3\end{array}\right) = \left(\begin{array}{c}1\\2\\3\end{array}\right) I'm not sure what these values are. Are the values...

Homework Statement Hey, I posted another question yesterday, and thanks to the kindness and brilliance of hall of ivy, I was able to solve it. However when I apply the same logic to this new question I cannot seem to get it, can someone explain or show me how to do this question. Consider the...

Homework Statement Consider the linear transformation T from V = P2 to W = P2 given by T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2 Let E = (e1, e2, e3) be the ordered basis in P2 given by e1(t) = 1, e2(t) = t, e3(t) = t^2 Find the coordinate matrix...

Homework Statement Prove that \dim L(\mathbb F)+\dim Ker L=\dim(\mathbb F+Ker L) for every subspace \mathbb{F} and every linear transformation L of a vector space V of a finite dimension. Homework Equations -Fundamental subspaces -Vector spaces The Attempt at a Solution Theorem: [/B]If...

Homework Statement Find all values a\in\mathbb{R} such that vector space V=P_2(x) is the sum of eigenvectors of linear transformation L: V\rightarrow V defined as L(u)(x)=(4+x)u(0)+(x-2)u'(x)+(1+3x+ax^2)u''(x). P_2(x) is the space of polynomials of order 2. Homework Equations -Eigenvalues and...

Homework Statement Find the general solution: http://puu.sh/ngck4/95470827b1.png Homework Equations Method: Gaussian Elimination by row operations. The Attempt at a Solution http://puu.sh/ngcml/7722bef842.jpg I am getting the wrong answer( w = -27/5). The solutions provided to me says the...

Homework Statement Determine whether the set spans ℜ3. If the set does not span ℜ3 give a geometric description of the subspace it does span. s = {(1, 0, 3), (2, 0, -1), (4, 0, 5), (2, 0, 6)} Homework EquationsThe Attempt at a Solution I am having trouble with the second part of this problem...

I'm looking for an excellent introductory linear algebra textbook for my second year pure mathematics course. My lecturer highly recommended Introduction to Linear Algebra by Marcus and Minc. She said she has searched for it for many years without success, as it is out of print. I love classic...

Hi all! I have an important decision to make for the summer of 2016 and I need some advice from some who have taken these courses. I need one biology lab elective to graduate, but it is a field lab and it runs from from 5/13 - 6/19. Because it is a field lab, I will not be able to take other...

Homework Statement find the general solution of the given system of equations: http://puu.sh/ncKaS/57a333f5b9.png Homework Equations Row Echelon Operations The Attempt at a Solution http://puu.sh/ncKcm/3e2b2bd5ab.jpg The correct answer given is x = 1, y = 1, z = 2, w = −3 I have done...

This might be a simple and pretty basic question, but i have not succeeded on finding any relevant info online, so hopefully someone can help me out. Is it possible to pull and actuator and it resists being pulled with a preset amount of force? What I'm thinking is e.x you have set a preset a...

Homework Statement Show that all square matrix (A whatever) can be written as the sum of a symmetric matrix and a anti symmetric matrix. Homework Equations I think this relation might be relevant : $$ A=\frac{1}{2}*(A+A^{T})+\frac{1}{2}*(A-A^{T}) $$ The Attempt at a Solution I know that we...

Hi guys, I'm having a debate with a mechanical engineer friend of mine, and I was wondering if you could help me solve it. I'm not much of a physicist, but honestly I think he might have this one wrong, I just can't remember my old physics classes well enough to calculate and be sure. The...

Homework Statement Rolling without slipping A) Derive the linear acceleration vector equations for points A, B, C, and O in terms of R, ω, α and θ at this instant. B) R = 0.5 m, ω=-54 r/s and α = 0. Determine the MPH of the vehicle and the vector accelerations of points A, B, C, and O. C) R...

I am revising the basics of linear transformations and trying to get a thorough understanding of linear transformations and their matrices ... ... At present I am working through examples and exercises in Seymour Lipshutz' book: Linear Algebra, Fourth Edition (Schaum Series) ... ... At...

Firstly, my apologies to Deveno in the event that he has already answered these questions in a previous post ... Now ... Suppose we have a linear transformation $$T: \mathbb{R}^3 \longrightarrow \mathbb{R}^2$$ , say ... Suppose also that $$\mathbb{R}^3$$ has basis $$B$$ and $$\mathbb{R}^2$$...

Homework Statement Check if L(p)(x)=(1+4x)p(x)+(x-x^2)p'(x)-(x^2+x^3)p''(x) is a linear transformation on \mathbb{R_2}[x]. If L(p)(x) is a linear transformation, find it's matrix in standard basis and check if L(p)(x) is invertible. If L(p)(x) is invertible, find the function rule of it's...

I have some high voltage (300v+) analog circuits that I want to control digitally which requires the use of a voltage controlled linear resistor that can withstand high voltages. I don't expect the current levels to be that high. I originally settled on LDR optocouplers but it turns out they...

As the discovery matches templates based on GR, and the regime is of very strong gravitational fields and very high speeds (relativistic speeds), and there is a 90% match between model and measured data, this does rule out linear or quasi linear alternative theories of gravity?

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... I am currently focussed on Chapter 2: Linear Algebra Essentials ... and in particular I am studying Section 2.6 Constructing Linear Transformations ... I need help with a basic...

Homework Statement [/B] Consider a long chain of mass m and length L suspended from a tall ceiling. Like any string if one end is disturbed waves will travel along the string. However, the tension in the string is due to its own weight and is not uniform. As such the speed of the wave will be...

Homework Statement Let U is the set of all commuting matrices with matrix A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}. Prove that U is the subspace of \mathbb{M_{3\times 3}} (space of matrices 3\times 3). Check if it contains span\{I,A,A^2,...\}. Find the...

Hi All, I am trying to understand better the tests used to determine the existence of a linear relation between two variables X,Y. AFAIK, one way of testing the strength of any linear relationship is by computing ##r^2##, where ##r## is the correlation coefficient; this measures the extend to...

In Andrew McInerney's book: First Steps in Differential Geometry, Theorem 2.4.3 reads as follows:https://www.physicsforums.com/attachments/5252McInerney leaves the proofs for the Theorem to the reader ... I am having trouble formulating a proof for Part (3) of the theorem ... Can someone help...

Two things I'd like to discuss: 1. The conservation of angular momentum. If you have two discs rotating on the same fixed rigid axis, will these nullify each other? I.e. Create no net angular momentum? 2. How / is it possible to convert angular momentum to linear momentum in the sense to be...

Guys, I'm trying to figure some stuff out, but I'm stumped. I need to figure the what my output would be. I'm applying linear force via hydraulic cylinder. The cylinder will turn a "gear" and in turn turn another. The "gear" is not necessarily a gear, it and the hydraulic cylinder will mate...

Homework Statement The image of a linear transformation = columnspace of the matrix associated to the linear transformation. More specifically though, given the transformation from Rn to Rm: from subspace X to subspace Y, the image of a linear transformation is equal to the set of vectors in X...

hi all i need a good Reference about mathematics my level in mathematics as zero

Homework Statement Two objects with mass of m are connected with a rod with length 2l and with no mass. The center of the rod is pinned so that it can spin. Object with mass M comes with speed v and sticks to m. There is no friction. 1) What is the angular speed w after collision? 2) FInd the...

Hi, While solving a system of linear equations, there are three possible cases - unique / infinite / no solutions - to the system. One geometric interpretation is when one looks at a set of planes intersecting at one / many / no points respectively, for each of the above cases. While going...

So I have NO IDEA how to do this problem. I am assuming it has to do with linear transport because that's the next section in the book but we have yet to talk about this in lab and briefly (one slide) discussed this in lecture. I tried reading in my book which was did no good as it just showed...

Decide which of the following equations are true or false. If false, explain/provide a counterexample. (a) If $a, b \in \mathbb{Z}$ are relatively prime, then $ax+by = N$ has integer solutions for any integer $N$. (b) The equation $70x+42y = 1409$ has integer solutions (c) The equation...

Common sense would suggest that all observed properties in a linear system would likely be linear. The response of a pendulum to an impulse can be computed, and for multiple impulses, the individual solutions can be superimposed to obtain the resultant solution. Qualitatively however...

Homework Statement If acceleration is not equal to 0, then which of the following must not be able to stay constant? I: Speed II: Linear Momentum III: Kinetic Energy (Select any 1, 2, or all of the above, or all of the above) Homework Equations KE = (1/2)mv^2 P = mv The Attempt at a Solution...

Homework Statement Being ##f : \mathbb R^4\rightarrow\mathbb R^4## the endomorphism defined by: $$f((x,y,z,t)) = (13x + y - 2z + 3t, 10y, 9z + 6t, 6z + 4t)$$ 1) Determine the basis and dimension of ##Ker(f)## and ##Im(f)##. Complete the base chosen in ##Ker(f)## into a base of ##\mathbb R^4##...

I'm looking for a nice set of basis matrices ##B_{i,j}## that cover the matrices of size ##n \times n## when linear combinations are allowed. The nice property I want them to satisfying is something like ##B_{i,j} \cdot B_{a,b} = B_{i+a, j+b}##, i.e. I want multiplication of two basis matrices...

Homework Statement How many points of intersection does the line y = x have with the parabola y = 6x − x2 −10 ? A) none B) one C) two D) three E) more than three Homework EquationsThe Attempt at a Solution x = 6x - x^2 - 10 -x^2+5x - 10 Used b^2 - 4ac to check the number of intersections. The...

So I am having difficulty with the following problem; Determine the currents in the various branches. So I went ahead and assigned I names to the various branches and drew in flow directions to help me visualize the problem better. From there I created the following three equations...

Homework Statement I can't understand this paper. I understand the whole incidence matrix stuff, but I don't quiet get how it relates to the linear algebra. I don't know if this is allowed to do, but I will ask you questions line by line, so basically you will read the paper with me explaining...

Homework Statement Find the span of U=\{2,\cos x,\sin x:x\in\mathbb{R}\} (U is the subset of a space of real functions) and V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\} Homework Equations - Span -Subset The Attempt at a Solution...

Homework Statement Question: What is the defect of a linear transformation? 2. The attempt at a solution A defective matrix (of a linear transformation) is a matrix that doesn't have a complete basis of eigenvectors. Does this mean that linearly dependent vectors of a matrix are called defects?