Linear algebra Definition and 999 Threads

Homework Statement Plot the solution set of linear equations x-y+2z-t=1 2x-3y-z+t=-1 x+7z=8 and check if the set is a vector space. 2. The attempt at a solution Augmented matrix of the system: \begin{bmatrix} 1 & -1 & 2 & -1 & 1 \\ 2 & -3 & -1 & 1 & -1 \\ 1 & 0 & 7 & 0 & 8 \\...

Homework Statement Prove that, if ##T,S\in \mathcal{L}(V)## then ##TS## and ##ST## have the same eigenvalues. Homework EquationsThe Attempt at a Solution Suppose ##T## is written in a basis in which its matrix is upper triangular, and so is ##S## (these bases may be of different list of...

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...

Homework Statement The question asks to show whether the following are sub-spaces of R^3. Here is the first problem. I want to make sure I'm on the right track. Problem: Show that W = {(x,y,z) : x,y,z ∈ ℝ; x = y + z} is a subspace of R^3. Homework Equations None The Attempt at a Solution...

During lecture, the professor gave us a theorem he wants us to prove on our own before he goes over the theorem in lecture. Theorem: Let ##V_1, V_2, ... V_n## be subspaces of a vector space ##V##. Then the following statements are equivalent. ##W=\sum V_i## is a direct sum. Decomposition of...

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 Let U is the set of all polynomials u on field \mathbb F such that u(3)=u(-2)=0. Check if U is the subspace of the set of all polynomials P(x) on \mathbb F and if it is, determine the set W such that P(x)=U\oplus W. Homework Equations -Polynomial vector spaces -Subspaces...

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 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 Homework Equations[/B] The Attempt at a Solution From that point, I don't know what to do. How do I prove linear independence if I have no numerical values? Thank you.

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...

Homework Statement Suppose that AB = AC for matrices A, B, and C. Is it true that B must equal C? Prove the result or find a counterexample. Homework Equations Properties of matrix multiplication The Attempt at a Solution AC = A(D + B) = AD + AB = 0 + AB = AB ? Can someone help me...

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...

Homework Statement Write in Vector-Matrix form then write the augmented matrix of the system. r + 2s + t = 1 r - 3s +3t = 1 4s - 5t = 3 Homework Equations The matrix to which the operations will be applied is called the augmented matrix of the system Ax = b, It is formed by appending the...

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, i now studying vector calculus, and for sheer curiosity i would like know if there exist a direct fashion to generalize the rotor operator, to more than 3 dimensions! On wiki there exist a voice https://en.wikipedia.org/wiki/Curl_(mathematics)#Generalizations , but I do not know how you...

Ok, so this is really weird. I am a second year physics major at a decent university. My GPA at the moment is not steller (3.3). I think I totally bombed my lower division linear algebra midterm, and I might (a big maybe, because I am confident that I could pass this class if I put my 110%...

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

Homework Statement If you have the heat equation $$u_{t}-u_{xx}=a \\ u(0,t)=b\\u(1,t)=c\\u(x,0)=d$$ Show that the solution to the above equation can be made up of a linear combination of solutions to $$u_{t}-u_{xx}=a_i \\ u(0,t)=b_i\\u(1,t)=c_i\\u(x,0)=d_i$$ $$i=1,2,3,4$$ if the following...

This is my question: What is the largest m such that there exist v1, ... ,vm ∈ ℝn such that for all i and j, if 1 ≤ i < j ≤ m, then ≤ vi⋅vj = 0 I found a couple of solutions online. http://mathoverflow.net/questions/31436/largest-number-of-vectors-with-pairwise-negative-dot-product...

Homework Statement In Zettili's QM textbook, we are asked to find the trace of an operator |\psi><\chi| . Where the kets |\psi> and |\chi> are equal to some (irrelevant, for the purposes of this question) linear combinations of two orthonormal basis kets. Homework 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?

Homework Statement Let and are two basis of subspaces and http://www.sosmath.com/CBB/latexrender/pictures/69691c7bdcc3ce6d5d8a1361f22d04ac.png. Find one basis of http://www.sosmath.com/CBB/latexrender/pictures/38d4e8e4669e784ae19bf38762e06045.png and...

I want a good linear algebra textbook in order to learn to use linear algebra in physics but also to use it in more theoretical mathematics courses. I hope that with this poll i will also help others that want to study from a proper Linear Algebra textbook.

Dear Physics Forum personnel, I am currently reading the books called "Linear Algebra Done Right" by S. Axler and "Linear Algebra Done Wrong" by S. Treil. On the next semester, I will be taking the "Second Course in Linear Algebra" which will treat the following topics: determinants...

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 -Vector space span -Linear independence...

I am reading Steven Roman's book, Advanced Linear Algebra and am currently focussed on Chapter 1: Vector Spaces ... ... I need help/clarification with respect to a notational issue regarding Roman's definition of the direct product and external direct sum of a family of vector spaces ... ...

Intro to Lin. Alg via MIT OCW 18.06 (part of Electrical Engineering), by Gilbert Strang. I've been doing it as independent study before starting my BSEE next year. I'm getting to the end. Chapter 9 is titled "Numerical Linear Algebra", and is concerned with the heavy, intricate computational...

Homework Statement Find basis and dimension of V,W,V\cap W,V+W where V=\{p\in\mathbb{R_4}(x):p^{'}(0) \wedge p(1)=p(0)=p(-1)\},W=\{p\in\mathbb{R_4}(x):p(1)=0\} Homework Equations -Vector spaces The Attempt at a Solution Could someone give a hint how to get general representation of a vector...

My Intro to LA course has visited the ideas of polar decomposition and Jordan forms, but not gone into them in depth. I wouldn't say I understood them, but I'm aware of them, and could possibly solve some basic exercises involving them if all I had to do was apply formulas. My question is...

Homework Statement 1) In a vector space V of all real polynomials of third degree or less find basis B such that for arbitrary polynomial p \in V the following applies: [p]_B = \begin{pmatrix} p'(0)\\p'(1)\\p(0)\\p(1)\end{pmatrix} where p' is the derivative of the polynomial p. Homework...

Homework Statement Check the statement is true or false: Let \mathcal{A} : \mathbb{R^3}\rightarrow \mathbb{R^4} be a linear operator. If the minimum rank of \mathcal{A} is 2, than the maximum defect is 1. Homework Equations -Linear transformations The Attempt at a Solution Assume that...

Homework Statement Let \phi:M_{2,2}\mathbb{(R)}\rightarrow \mathcal{P_2} be a linear operator defined as: (\phi(A))(x)=tr(AB+BA)+tr(AB-BA)x+tr(A+A^T)x^2 where B= \begin{bmatrix} 3 & -2 \\ 2 & -2 \\ \end{bmatrix} Find rank,defect and one basis of an image and kernel of linear operator...

Apologies if this is a really trivial question, but I've never been quite sure as to the usage of the terminology dual space. I get that given a vector space ##V## we can construct a set of linear functionals that map ##V## into its underlying field and that these linear functionals themselves...

Homework Statement $$\dot{x_1}=x_2-x_2^3,~~~~~~\dot{x_2}=-x_1-3x_2^2+x_1^2x_2+x_2$$ I need help in determining the type and stability of the fixed points in this system. Homework Equations The Jordan Normal Form[/B] Let A be a 2x2 matrix, then there exists a real and non singular matrix M...

Hey guys, I'm talking to my advisor soon and I was wondering if it is typical to, after taking multi variable calc, to take differential equations and linear algebra simultaneously? I'm going to have to be taking both modern physics and organic chemistry II as well, for context. Thanks everyone

Hello good people, please refer to this: (notice the mistake in 9.31: cos(psi) switches places with cos(phi)sin(psi) to the best of my understanding) Now, I am trying to derive 9.30 and for this, according to the book, we solve 9.32. The problem is I can not understand 9.32, the meaning of...

Homework Statement Let T: P2 --> P3 be the transformation that maps a polynomial p(t) into the polynomial (t+5)p(t). a) find the image of p(t)= 2-t+(t^2) b) Find the matrix for T relative to bases {1,t,t^2} and {1,t,t^2,t^3}. Homework Equations Given The Attempt at a Solution a) I know...

Homework Statement What books of completely solved problems (free in pdf) in linear algebra would you suggest? Please suggest books that have solved problems, and not theory. Homework EquationsThe Attempt at a Solution

Given a Positive Definite Matrix ## A \in {\mathbb{R}}^{2 \times 2} ## given by: $$ A = \begin{bmatrix} {A}_{11} & {A}_{12} \\ {A}_{12} & {A}_{22} \end{bmatrix} $$ And a Matrix ## B ## Given by: $$ B = \begin{bmatrix} \frac{1}{\sqrt{{A}_{11}}} & 0 \\ 0 & \frac{1}{\sqrt{{A}_{22}}}...

Homework Statement [/B] If A is an mxn matrix, show that for each invertible nxn matrix V, im(A) = im(AV) Homework Equations none The Attempt at a Solution I know that im(A) can also be written as the span of columns of A. I also know that AV = [Av1 Av2 ... Avn] so im(AV) is the span of...

It is from some famous publications. But I seem can get it from rigorous proof after many hours and different methods of trying and Googling. If we have g as the eigenvector of a symmetric matrix and G is the eigenvalue of the symmetric matrix. \left[ {\begin{array}{*{20}{c}} {X11 -...

In my circuit analysis class I consistently need to solve system of complex equations, and I can't use MATLAB or anything for it. Suppose I have the following system: (Va-Vs)/(-j15) + Va/33 + (Va-Vo)/(-j25)=0 (Vo-Va)/(-j25) + (Vo-Vs)/10 = 0 What is the best way to solve it by hand in a time...