Linear algebra Definition and 999 Threads

Homework Statement Consider hermitian matrices M1, M2, M3, M4 that obey the property Mi Mj + Mj Mi = 2δij I where I is the identity matrix and i,j=1,2,3,4 a) Show that the eigenvalues of Mi=+/- 1 (Hint: Go to the eigenbasis of Mi and use the equation for i=j) b) By considering the relation Mi...

I have to find a unitary transformation that takes me from one quantum state to another (or if there is such a transformation), given the two quantum states in matrix form. The matrices are huge (smallest is 16x16) , so doing it on paper is not an option. Does anyone know how I can do this in...

I am trying to solve the following problem: Let A be a real mxn matrix. Describe the set of all vectors in F^m orthogonal to Col(A). Here, F^m could be C^m. Now in the real case, I'd say that the column space of A is the row space of A^T, and it is well known that the row space of a matrix is...

Homework Statement I'm reading Zee's book Einstein Gravity, I'm in the section where he said that given an array of two numbers p=(ap1, bp2), it is not a vector unless a=b. He just stated it without really showing how it must be like that. I know that a vector should satisfy a transformation...

In Bruce Cooperstein's book: Advanced Linear Algebra, he gives the following example on page 12 in his chapter on vector spaces (Chapter 1) ... ...I am finding it difficult to fully understand this example ... ... Can someone give an example using Cooperstein's construction ... using, for...

I'm grading for a linear algebra class this semester. The class is comprised entirely of engineering majors of various flavors. The homework assigned by the professor is almost entirely "proofs" they are fairly specific proofs. Really the only thing that designates them as proofs is that the...

This question mostly pertains to how looking at affine independence entirely in terms of linear independence between different families of vectors. I understand there are quite a few questions already online pertaining to the affine/linear independence relationship, but I'm not quite able to...

Hi, I am looking for a book that explains tensors and builds a working knowledge of tensors, like the book Linear Algebra by Friedberg Insel and Spence, which I thought explained things very well (if you haven't heard of it, its an intro. book on linear algebra). Thanks!

I hope this post won't become too tedious. I've completed my undergrad studies in physics and if things go well i will begin my master's degree in April. The thing is, since my path to graduation has been peculiar (to say the least) I'm kinda weak in maths skills atm and need to improve. I'm...

Homework Statement First, I'd like to say that this question is from an Introductory Linear Algebra course so my knowledge of vector space and subspace is limited. Now onto the question. Q: Which of the following are subspaces of F(-∞,∞)? (a) All functions f in F(-∞,∞) for which f(0) = 0...

Hi, I'm currently studying to become a chemical engineer. After learning differential equation and linear algebra, I've realized how useful they are in my engineering courses since they make setting up equations and solving them so much easier. So I was wondering if there are other math that...

Homework Statement Construct a 3x3 matrix A and vectors b and c in R^3 so that Ax=b has a solution but Ax=c Homework EquationsThe Attempt at a Solution So I don't know where to start. I am not sure if the problem is asking me to create a matrix with real numbers or variables. What I do know is...

1. The problem statement Given a stochastic matrix P with states s_1...s_5: P = \begin{pmatrix} 1 & p_2 & 0 & 0 & 0\\ 0 & 0 & p_3 & 0 & 0\\ 0 & q_2 & 0 & p_4 & 0\\ 0 & 0 & q_3 & 0 & 0 \\ 0 & 0 & 0 & q_4 & 1 \end{pmatrix} and the matrix A (which is obviously related to P, but I can't see...

Homework Statement Let A be an nxn matrix belonging to R and x be a vector of length k belonging to R. Find the first column of M = (A − x1I)(A − x2I)...(A − xrI) using a sequence of GAXPY’s operations. Homework Equations GAXPY: General matrix A multiplied by a vector X plus a vector Y The...

Homework Statement Prove that if ({A_1, A_2, ..., A_k}) is a linearly independent subset of M_nxn(F), then (A_1^T,A_2^T,...,A_k^T) is also linearly independent. Homework EquationsThe Attempt at a Solution Have: a_1A_1^T+a_2A_2^T+...+a_kA_k^T=0 implies a_1A_1+a_2A_2+...+a_kA_k=0 So...

Homework Statement This is a homework problem for my Honors Calculus I class. The problem I'm having is that though I can solve a traditional function composition problem, I'm stumped as to how to do this for multivariate functions. I read that it requires an extension of the notion of...

Homework Statement Determine if the following sets are bases for P_2(R) b) (1+2x+x^2, 3+x^2,x+x^2) d) (-1+2x+4x^2, 3-4x-10x^2,-2-5x-6x^2) Homework Equations Bases IF Linear Independence AND span(Set)=P_2(R) RREF = Reduced Row Echelon Form The Attempt at a Solution My first question here...

Dear all, Can someone please recommend a english version of a linear algebra book. "Lineare algebra, Siegfried Bosch" I don't speak German so I cannot understand the chapters. If someone can help to identify a book with similar chapters that will be very helpful. you can find the full book...

Suppose we pick a matrix M\in M_n(ℝ) s.t. all its eigenvalues are strictly bigger than 1. In the question here the user said it induces some norm (|||⋅|||) which "expands" vector in sense that exists constant c∈ℝ s.t. ∀x∈ℝ^n |||Ax||| ≥ |||x||| . I still cannot understand why it's correct. How...

Homework Statement True or false? If T: ℙ8(ℝ) → ℙ8(ℝ) is defined by T(p) = p', so exists a basis of ℙ8(ℝ) such that the matrix of T in relation to this basis is inversible. Homework EquationsThe Attempt at a Solution So i think that my equations is of the form: A.x = x' hence A is...

What are some books that I can use master DiffyQ and linear algebra ? I want to improve significantly in E&M and quantum mechanics. I understand that I have to practice, practice, and practice , but can you provided me with with some advice ? Thank you.

It's from the chapter on Matrix Inverses... imgur link: http://i.imgur.com/8OhFzgi.jpg This is the entirety of the exercise. It's not following on from or setting anything else up. That's just number 42...what can I do with this?

Course: MIT OCW 18.06 Intro to Linear Algebra by Strang 4th edt. Question: if a3,3 is 7, and the third pivot is 5, if we change a3,3 to be 11, then the third pivot becomes _________. If you change the a3,3 to ________ then there is no third pivot. At first I thought a3,3 had to be the actual...

Homework Statement Prove that a set of linearly independent vectors in Rn can have maximum n elements. So how would you prove that the maximum number of independent vectors in Rn is n?I can understand why in my head but not sure how to give a mathematical proof. I understand it in terms of the...

Hello all, I am about a week into a quantum mechanics course in which the instructor is mostly going to follow along with Sakurai's Modern Quantum Mechanics. However, my linear algebra is pretty rusty and I have never taken an actual course in linear algebra, I have just learned what I have...

Homework Statement Suppose A is non-singular (nxn) matrix. Given that AB=(zero matrix), show that B=(zero matrix). Hint: Express A as [A1, A2, ..., An] and Express AB as [AB1, AB2, ..., ABn] Homework EquationsThe Attempt at a Solution [/B] I argued that because A is non-singular, A=[A1...

Dear Physics Forum personnel, I am curious if the euclidean space R^n is an example of vector space. Also can matrices with 1x2 or 2x1 dimension be a vector for the R^n? PK

I feel like I almost understand the solution I've come up with, but a step in the logic is missing. I'll post the question and my solution in LaTeX form. Paraphrasing of text question below in LaTeX. Text question can be seen in its entirety via this imgur link: http://i.imgur.com/41fvDRN.jpg...

Hi, rapid fire posting in this subforum I know, sorry if that's annoying. Let me know if I should space my posts out a bit more. Here's an image of the solution to a worked example (from Intro to Linear Algebra 4th by Strang) here's the imgur link: http://i.imgur.com/IG6r15H.jpg I cannot...

MIT OCW 18.06 using Intro to Linear Algebra by Strang So I was working through some stuff about Cyclic Matrices, and the text was talking about how the column vectors that make up this cyclic matrix, shown here, are coplanar, and that is the reason that Ax = b will have either infinite...

Doing MIT OCW 18.06 using Gilbert Strang Intro to Linear Algebra. Ch 1.2 The vectors that are perpendicular to <1,1,1> and <1,2,3> lie on a ___________. I would have said "plane". I've worked with vectors in calculus, and if you take the cross product of those two vectors you get a vector...

I am preparing myself for a BSEE starting next year. I have just finished Stewarts Calculus for Calc I,II,III and Diff Eqs up to solving 2nd order non-hom diff eqs by undet. coeffs, variation of parameters and power series. In high school some 15 years ago I saw some very basic linear algebra...

How can I find the parametric vector form of a cartesian equation under a specific condition? Cartestian equation: $$-2x-y+z=6$$ I know to find the parametric vector form we can find any 3 points P, Q and R which satisfy the cartesian equation. $$ \begin{pmatrix} x_1\\ y_1\\ z_1...

Homework Statement Suppose a linear transformation T: [P][/2]→[R][/3] is defined by T(1+x)= (1,3,1), T(1-x)= (-1,1,1) and T(1-[x][/2])=(-1,2,0) a) use the given values of T and linearity properties to find T(1), T(x) and T([x][/2]) b) Find the matrix representation of T (relative to standard...

Homework Statement Let B1={([u][/1]),([u][/2]),([u][/3])}={(1,1,1),(0,2,-1),(1,0,2)} and B2={([v][/1]),([v][/2]),([v][/3])}={(1,0,1),(1,-1,2),(0,2,1)} a) Show that B1 is a basis for [R][/3] b) Find the coordinates of w=(2,3,1) relative to B1 c)Given that B2 is a basis for [R[/3], find...

Homework Statement What is the volume of the parallelepiped with vertices (3, 0, −1), (4, 2, −1), (−1, 1, 0), (3, 1, 5), (0, 3, 0), (4, 3, 5), (−1, 2, 6), and (0, 4, 6)? Homework Equations The volume of a parallelepiped is given by the triple scalar product, (a × b) ⋅ c The Attempt at a...

Hi, I want to learn Linear Algebra in its most rigorous and expansive form. I have narrowed down to two books (well, one is a series). On one hand, I want to try Linear Algebra by Hoffman/Kunze, but my school's library has Lang's Introduction to Linear Algebra, and his second book on the...

Homework Statement Let V be the set of all ordered pairs of real numbers. Suppose we define addition and scalar multiplication of elements of V in an unusual way so that when u=(x1, y1), v=(x2, y2) and k∈ℝ u+v= (x1⋅x2, y1+y2) and k⋅u=(x1/k, y1/k) Show detailed calculations of one case...

[FONT=Times New Roman]Homework Statement From Linear Algebra with applications 7th Edition by Keith Nicholson. Chapter 9.2 Example 2. Let T: R3 → R3 be defined by T(a,b,c) = (2a-b,b+c,c-3a). If B0 denotes the standard basis of R3 and B = {(1,1,0),(1,0,1),(0,1,0)}, find an invertible matrix P...

Hey all. So I have been reading this article and have a question I would like to ask. I will be referring to this article extensively so it would be kind of you to open it: http://www.ee.ucr.edu/~yhua/MILCOM_2013_Reprint.pdf I believe reading the article is not required to answer my questions...

Homework Statement Let T:V→V be a linear operator on a vector space V over C: (a) Give an example of an operator T:C^2→C^2 such that R(T)∩N(T)={0} but T is not a projection (b) Find a formula for a linear operator T:C^3→C^3 over C such that T is a projection with R(T)=span{(1,1,1)} and...

I just finished up Stewart's Calculus Textbook, and the last section was on solving 2nd Order Non-Homogeneous Diff Eqs using power series. I've looked through Paul's Calculus page in the Differential Sections, and can see that there is still a lot more beyond Stewart's that I'd like to study...

1. Get A from its inverse3. I believe it has something to do with the theorem that states: E1E2E3...EkA=I

What are some rigorous theoretical books on mathematics for each branch of it? I have devised a fantastic list of my own and would like to hear your sentiments too. Elementary Algebra: Gelfand's Algebra Gelfand's Functions & Graphs Burnside's Theory of Equations Euler's Analysis of the...

I want to know what exactly Eigen value imply. What is its Physical significance ? Physical significance of eigen vector? Does eigen value concept apply in signal processing or evalvating frequency response off a system?

Does anyone know of any gentle, introductory books to LA that assume little prerequisites, even in the way of vectors and matrices? I want something that will give intuition and reasonable proofs, and will provide enough background for something like computational neuroscience. I do not know...

In my school LA requires a pre req proof class vs matrix algebra which doesn't. Would matrix algebra even be worth taking?

Homework Statement Determine whether or not the vector functions are linearly dependent? u=(2t-1,-t) , v= (-t+1,2t) and they are written as columns matrixes. Homework Equations Wronskian, but I don't think I should use it because I need to take derivatives so it doesn't seem like it would...

Construct a linear system determined by four numbers whose sum is 40, with the first three numbers adding up to 20 and the last three to 30. a) Explain why this system has infinitely many solutions. b) Add another condition on the numbers so that a unique solution can be found and then find...

Dear Physics Forum advisers, I am a rising college junior in U.S. with a major in mathematics, and an aspiring applied mathematician. I apologize for this sudden interruption, but I wrote this email to seek your advice on my current problem on the course selection. I will very soon be...