Linear algebra Definition and 999 Threads

Hi physicsforums, I am an undergrad currently taking an upper-division course in Quantum Mechanics and we have begun studying L^2 space, state vectors, bra-ket notation, and operators, etc. I have a few questions about the relationship between L^2, the space of square-integrable complex-valued...

So, I recently came across this example: let us "define" a function as ƒ(x)=-x3-2x -3. If given a matrix A, compute ƒ(A). The soution proceedes in finding -A3-2A-3I where I is the multiplicative identity matrix. Now , I understand that you can't add a scalar and a matrix, so the way I see it is...

Homework Statement Suppose that A is a 3 x n matrix whose columns span R3. Explain how to construct an n x 3 matrix D such that AD = I3. "Theorem 4" For a matrix A of size m x n, the following statements are equivalent, that is either all true or all false: a. For each b in Rm, Ax = b has a...

Given an ##N## dimensional binary vector ##\mathbf{v}## whose conversion to decimal is equal to ##j##, is there a way to linearly map the vector ##\mathbf{v}## to an ##{2^N}## dimensional binary vector ##\mathbf{e}## whose ##(j+1)##-th element is equal to ##1## (assuming the index starts...

The question comes out of a corollary of this theorem: Let B be a symmetric bilinear form on a vector space, V, over a field \mathbb{F}= \mathbb{R} or \mathbb{F}= \mathbb{C}. Then there exists a basis v_{1},\dots, v_{n} such that B(v_{i},v_{j}) = 0 for i\neq j and such that for all...

Post moved by moderator, so missing the homework template. series ##{a_n}## is define by ##a_1=1 ## , ##a_2=5 ## , ##a_3=1 ##, ##a_{n+3}=a_{n+2}+4a_{n+1}-4a_n ## ( ##n \geq 1 ##). $$\begin{pmatrix}a_{n+3} \\ a_{n+2} \\ a_{n+1} \\ \end{pmatrix}=B\begin{pmatrix}a_{n+2} \\ a_{n+1} \\ a_{n} \\...

Homework Statement Find the Fourier series of the function ##f## given by ##f(x) = 1##, ##|x| \geq \frac{\pi}{2}## and ##f(x) = 0##, ##|x| \leq \frac{\pi}{2}## over the interval ##[-\pi, \pi]##. Homework Equations From my lecture notes, the Fourier series is ##f(t) = \frac{a_0}{2}*1 +...

<Moderator's note: Moved from a technical forum and thus no template.> Hello I am given the following problem to solve. Identify the quadratic form given by ##-5x^2 + y^2 - z^2 + 4xy + 6xz = 5##. Finally, plot it. I cannot seem to understand what I have to do. The textbook chapter on...

Homework Statement How to prove ##max\{0, \rho(\sigma)+\rho(\tau)-m\}\leq \rho(\tau\sigma)\leq min\{\rho(\tau), \rho(\sigma)\}##? Homework Equations Let ##\sigma:U\rightarrow V## and ##\tau:V\rightarrow W## such that ##dimU=n##, ##dimV=m##. Define ##v(\tau)## to be the nullity of ##\tau##...

Homework Statement Homework Equations I am not sure. I have not seen the triangle inequality for inner products, nor the Cauchy-Schwarz Inequality for the inner product. The only thing that my lecture notes and textbook show is the axioms for general inner products, the definition of norm...

A problem that I have to solve for my Linear Algebra course is the following We are supposed to use Mathematica. What I have done is that I first checked that A is symmetric, i.e. that ##A = A^T##. Which is obvious. Next I computed the eigenvalues for A. The characteristic polynomial is...

I am trying to understand the geometric intuition of the above equation. ##\rho(\tau)## represents the rank of the linear transformation ##\tau## and likewise for ##\rho(\tau\sigma)##. ##Im(\sigma)## means the image of the linear transformation ##\sigma## and lastly, ##K(\tau)## is the kernel of...

Homework Statement Let ##W_1=\langle (1,2,3,6),(4,-1,3,6)(5,1,6,12))\rangle## and ##W_2=\langle (1,-1,1,1),(2,-1,4,5)\rangle## be subspaces of ##\Bbb{R}^4##. Find the bases for ##W_1\cap W_2## and ##W_1+W_2##. Homework Equations...

Homework Statement Let A be an arbitrary m× n matrix. Find a matrix C such that CA = B for each of the following matrices B. a. B is the matrix that results from multiplying row i of A by a nonzero number c. b. B is the matrix that results from swapping rows i and j of A. c. B is the matrix...

Homework Statement Find a spanning set for ##P_4##. Find a minimal spanning set. Use Theorem 2.7 to show no other spanning set has fewer elements. Would simply like someone to check my answers as the book I'm using did not provide a solutions manual. Thank you. Homework Equations Theorem 2.7...

Homework Statement [/B] Suppose the solution set of some system Ax = b , Where A is a 4x3 matrix, is *Bold characters are vectors* x_1= 1 + 3t x_2 = 2 - t x_3 = 3 + 2t where t is a parameter and can be any number. a) How many pivots are in the row echelon form of A? b) Let u, v, w be the...

I have not really finished studying linear algebra, I have to admit. The furthest I have gotten to is manipulating matrices a little bit (although I have used this in differential equations to calculate a Wronskian to see if two equations are linear independent, but again, a determinant is...

Let L1 be the line passing through the point P1=(−2,−11,9) with direction vector d2=[0,2,−2]T, and let L2 be the line passing through the point P2=(−2,−1,11) with direction vector d2=[−1,0,−1]T Find the shortest distance d between these two lines, and find a point Q1 on L1 and a point Q2 on L2...

Hi all. So to start I'll say I'm just dealing with functions of a real variable. In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects" So with that in mind, is there any geometric connection between two orthoganal functions on some...

I have the following matrix given by a basis \left|1\right\rangle and \left|2\right\rangle: \begin{bmatrix} E_0 &-A \\ -A & E_0 \end{bmatrix} Eventually I found the matrix eigenvalues E_I=E_0-A and E_{II}=E_0+A and eigenvectors \left|I\right\rangle = \begin{bmatrix} \frac{1}{\sqrt{2}}\\...

Homework Statement Show that S ⊆ T, where S and T are both subsets of R^3. Homework Equations S = {(1, 2, 1), (1, 1, 2)}, T ={(x,y,3x−y): x,y∈R} The Attempt at a Solution I considered finding if S is a spanning set for T but I'm aware that this is perhaps not relevant. If I find {α(1, 2, 1)...

Homework Statement I have an assignment for my linear algebra class, that I simply cannot figure out. Its going to be hard to follow the template of the forum, as its a rather simply problem. It is as follows: Given the following subspace (F = reals and complex) and the "linear image"...

Homework Statement Given the six vectors below: 1. Find the largest number of linearly independent vectors among these. Be sure to carefully describe how you would go about doing so before you start the computation. 2 .Let the 6 vectors form the columns of a matrix A. Find the dimension of...

Let us assume that d is a vector in the vector space ℝ2 , then is: {td | t ∈ ℝ} the same as span{d} ? Thank you.

Homework Statement The Hamiltonian of a certain two-level system is: $$\hat H = \epsilon (|1 \rangle \langle 1 | - |2 \rangle \langle 2 | + |1 \rangle \langle 2 | + |2 \rangle \langle 1 |)$$ Where ##|1 \rangle, |2 \rangle## is an orthonormal basis and ##\epsilon## is a number with units of...

Homework Statement (i) Reduce the system to echelon form C|d (ii) For k = -12, what are the ranks of C and C|d? Find the solution in vector form if the system is consistent. (iii) Repeat part (b) above for k = −18 Homework Equations Gaussian elimination I used here...

Homework Statement Given the following matrix: I need to determine the conditions for b1, b2, and b3 to make the system consistent. In addition, I need to check if the system is consistent when: a) b1 = 1, b2 = 1, b3 = 3 b) b1 = 1, b2 = 0., b3 = -1 c) b1 = 1, b2 = 2, b3 = 3 Homework...

Homework Statement Given this matrix: I am asked to find values of the coefficient of the second value of the third row that would make it impossible to proceed and make elimination break down. Homework Equations Gaussian elimination methods I used given here...

<Moved from a homework forum. Template removed.> I can't find any documentation on how to do this. I remember in linear algebra how to find the incidence matrix of an electrical network of purely resistors. Put how do I find it of a RLC circuit with resistors, inductors, and capacitors? I can't...

Homework Statement Verify that A^2-2A+7I=0Homework Equations A is a squared matrix and I is the identity matrix. The Attempt at a Solution I squared a matrix, which I called A, by multiplying the two A matrices together, then I subtracting the new matrix with the third matrix 2A, then I added...

hey everyone just started university and the jump i feel is huge from a level and was just wondering if you guys knew of any books that had linear algebra and/or several variable calculus in them but displayed and explained stuff in a clear simple way? or if anyone has any websites that do the...

I've started self-studying quantum mechanics. It's clear from google searching and online Q.Mech lectures, I'll need to understand linear algebra first. I'm starting with finite-dimensional linear algebra and Hoffman, Kunze is one of the widely recommended textbooks for that. I need help...

Homework Statement Determine the values of a for which the following system of linear equations has no solutions, a unique solution, or infinitely many solutions. You can select 'always', 'never', 'a = ', or 'a ≠', then specify a value or comma-separated list of values. x1−2x2+2x3 = −1...

I am fairly new here so I apologize for any mistakes in my post. My question concerning solving a system of equations using Gauss-Jordan Elimination is specifically about different ways to handle a possible constant. Say for instance you have three equations: X1+X2+X3 + 3 = 9 2X1+4X2+X3 =...

I am going to have two slots available this year for electives and I want to use one of them for Astronomy. For the other, I am struggling to decide between Linear Algebra or Computer Science (CIS 210 at my university) which focuses on Python programming. If I can only choose one, which is more...

Hello, everybody! I would really appreciate if someone could help me understand how to solve the following two tasks. I am not sure whether my translation is correct, so if, by any chance, you know a more appropriate terminology, please let me know. I am not fluent in writing matrices here on...

1. The problem statement, all variables, and given/known data Triangle ABC has a point D on the line segment AB which cuts the segment in ratio AD : DB = 2 : 1. Another point E is on the line segment BC, cutting it in ratio BE : EC = 1 : 4. Point F is the intersection of the line segments AE and...

It is the demonstration of an important theorem I do not succeed in understanding. "A matrix has rank k if - and only if - it has k rows - and k columns - linearly independent, whilst each one of the remaining rows - and columns - is a linear combination of the k preceding ones". Let's suppose...

Homework Statement The volume of a parallelepiped defined by the vectors w, u, \text{ and }v, \text{ where } w=u \times v is computed using: V = w \cdot (u \times v) However, if the parallelepiped is defined by the vectors w-u, u, \text{ and }v, \text{ where } w=u \times v instead, the volume...

Homework Statement We can treat the following coupled system of differential equations as an eigenvalue problem: ## 2 \frac{dy_1}{dt} = 2f_1 - 3y_1 + y_2 ## ## 2\frac{dy_2}{dt} = 2f_2 + y_1 -3y_2 ## ## \frac{dy_3}{dt} = f_3 - 4y_3 ## where f1, f2 and f3 is a set of time-dependent sources, and...

I know both are different courses, but what I mean is, will a proof based Linear Algebra course be similar to an Abstract Algebra course in terms of difficulty and proofs, or are the proofs similar? Someone told me that there isn't that much difference between the proofs in Linear or Abstract...

Homework Statement If A is an n x n matrix over R such that A^3 = A + 1, prove that det(A) > 0 . Homework EquationsThe Attempt at a Solution So, what I've done is factor the expression to get A(A+1)(A-1) = 1, then taking the determinant of both sides, I get det(A)det(A+1)det(A-1) = 1. I...

Homework Statement [/B] The trace of a matrix is defined to be the sum of its diaganol matrix elements. 1. Show that Tr(ΩΛ) = Tr(ΩΛ) 2. Show that Tr(ΩΛθ) = Tr(θΩΛ) = Tr(ΛθΩ) (the permutations are cyclic) my note: the cross here U[+][/+]is supposed to signify the adjoint of the unitary matrix U...

In the book, Introduction to Linear Algebra, Gilbert Strang says that every time we see a space of vectors, the zero vector will be included in it. I reckon that this is only the case if the plane passes through the origin. Else wise, how can a space contain a zero vector if it does not pass...

[mentor note: thread moved from Linear Algebra to here hence no homework template] So, i was doing a Linear Algebra exercise on my book, and thought about this. We have a linear map A:E→E, where E=C°(ℝ), the vector space of all continuous functions. Let's suppose that Aƒ= x∫0 ƒ(t)dt. By the...

I'm having trouble understanding a step in a proof about bilinear forms Let ## \mathbb{F}:\,\mathbb{R}^{n}\times\mathbb{R}^{n}\to \mathbb{R}## be a bilinear functional. ##x,y\in\mathbb{R}^{n}## ##x=\sum\limits^{n}_{i=0}\,x_{i}e_{i}## ##y=\sum\limits^{n}_{j=0}\;y_{j}e_{j}##...

Hey all, I'm currently working on my CS degree with a mathematics minor. After this Fall, I will only have one more course to take to finish my minor. I'm debating between Linear Algebra and Deterministic Operations Research. I do have other options, but these seem to be most applicable to CS...

Homework Statement Show that if T is normal, then T and T* have the same kernel and the same image. Homework Equations N/A The Attempt at a Solution At first I tried proving that Ker T ⊆ Ker T* and Ker T* ⊆ Ker, thus proving Ker T = Ker T* and doing the same thing with I am T, but could not...

Homework Statement Prove that a 2x2 complex matrix ##A=\begin{bmatrix} a & b \\ c & d\end{bmatrix}## is positive if and only if (i) ##A=A*##. and (ii) ##a, d## and ##\left| A \right| = ad-bc## Homework Equations N/A The Attempt at a Solution I got stuck at the first part. if ##A## is positive...

Homework Statement The problem relates to a proof of a previous statement, so I shall present it first: "Suppose P is a self-adjoint operator on an inner product space V and ##\langle P(u),u \rangle## ≥ 0 for every u ∈ V, prove P=T2 for some self-adjoint operator T. Because P is self-adjoint...